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TeX (original user input): 

\begin{align}
  & \Delta r=c\Delta t\to {{\left( \Delta r \right)}^{2}}={{c}^{2}}{{\left( \Delta t \right)}^{2}} \\
 & \to {{\left( \Delta {{x}^{1}} \right)}^{2}}+{{\left( \Delta {{x}^{2}} \right)}^{2}}+{{\left( \Delta {{x}^{3}} \right)}^{2}}={{c}^{2}}\Delta {{t}^{2}} \\
 & \to \Delta {{s}^{2}}:={{c}^{2}}\Delta {{t}^{2}}-{{\left( \Delta {{x}^{1}} \right)}^{2}}-{{\left( \Delta {{x}^{2}} \right)}^{2}}-{{\left( \Delta {{x}^{3}} \right)}^{2}} \\
 & \to d{{s}^{2}}={{c}^{2}}d{{t}^{2}}-{{\left( d{{x}^{1}} \right)}^{2}}-{{\left( d{{x}^{2}} \right)}^{2}}-{{\left( d{{x}^{3}} \right)}^{2}}=0 \\
\end{align}

TeX (checked): 

{\begin{aligned}&\Delta r=c\Delta t\to {{\left(\Delta r\right)}^{2}}={{c}^{2}}{{\left(\Delta t\right)}^{2}}\\&\to {{\left(\Delta {{x}^{1}}\right)}^{2}}+{{\left(\Delta {{x}^{2}}\right)}^{2}}+{{\left(\Delta {{x}^{3}}\right)}^{2}}={{c}^{2}}\Delta {{t}^{2}}\\&\to \Delta {{s}^{2}}:={{c}^{2}}\Delta {{t}^{2}}-{{\left(\Delta {{x}^{1}}\right)}^{2}}-{{\left(\Delta {{x}^{2}}\right)}^{2}}-{{\left(\Delta {{x}^{3}}\right)}^{2}}\\&\to d{{s}^{2}}={{c}^{2}}d{{t}^{2}}-{{\left(d{{x}^{1}}\right)}^{2}}-{{\left(d{{x}^{2}}\right)}^{2}}-{{\left(d{{x}^{3}}\right)}^{2}}=0\\\end{aligned}}

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MathML (45.123 KB / 4.425 KB) :

Δ ⁢ r = c ⁢ Δ ⁢ t → ( Δ ⁢ r ) 2 = c 2 ⁢ ( Δ ⁢ t ) 2 → ( Δ ⁢ x 1 ) 2 + ( Δ ⁢ x 2 ) 2 + ( Δ ⁢ x 3 ) 2 = c 2 ⁢ Δ ⁢ t 2 → Δ ⁢ s 2 := c 2 ⁢ Δ ⁢ t 2 - ( Δ ⁢ x 1 ) 2 - ( Δ ⁢ x 2 ) 2 - ( Δ ⁢ x 3 ) 2 → d ⁢ s 2 = c 2 ⁢ d ⁢ t 2 - ( d ⁢ x 1 ) 2 - ( d ⁢ x 2 ) 2 - ( d ⁢ x 3 ) 2 = 0 missing-subexpression Δ 𝑟 𝑐 Δ 𝑡 → superscript Δ 𝑟 2 superscript 𝑐 2 superscript Δ 𝑡 2 missing-subexpression → absent superscript Δ superscript 𝑥 1 2 superscript Δ superscript 𝑥 2 2 superscript Δ superscript 𝑥 3 2 superscript 𝑐 2 Δ superscript 𝑡 2 missing-subexpression → absent Δ superscript 𝑠 2 assign superscript 𝑐 2 Δ superscript 𝑡 2 superscript Δ superscript 𝑥 1 2 superscript Δ superscript 𝑥 2 2 superscript Δ superscript 𝑥 3 2 missing-subexpression → absent 𝑑 superscript 𝑠 2 superscript 𝑐 2 𝑑 superscript 𝑡 2 superscript 𝑑 superscript 𝑥 1 2 superscript 𝑑 superscript 𝑥 2 2 superscript 𝑑 superscript 𝑥 3 2 0 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle\Delta r=c\Delta t% \to{{\left(\Delta r\right)}^{2}}={{c}^{2}}{{\left(\Delta t\right)}^{2}}\\ &\displaystyle\to{{\left(\Delta{{x}^{1}}\right)}^{2}}+{{\left(\Delta{{x}^{2}}% \right)}^{2}}+{{\left(\Delta{{x}^{3}}\right)}^{2}}={{c}^{2}}\Delta{{t}^{2}}\\ &\displaystyle\to\Delta{{s}^{2}}:={{c}^{2}}\Delta{{t}^{2}}-{{\left(\Delta{{x}^% {1}}\right)}^{2}}-{{\left(\Delta{{x}^{2}}\right)}^{2}}-{{\left(\Delta{{x}^{3}}% \right)}^{2}}\\ &\displaystyle\to d{{s}^{2}}={{c}^{2}}d{{t}^{2}}-{{\left(d{{x}^{1}}\right)}^{2% }}-{{\left(d{{x}^{2}}\right)}^{2}}-{{\left(d{{x}^{3}}\right)}^{2}}=0\\ \end{aligned}}}

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MathML (4.502 KB / 467 B) :

Δr=cΔt→(Δr)2=c2(Δt)2→(Δx1)2+(Δx2)2+(Δx3)2=c2Δt2→Δs2:=c2Δt2−(Δx1)2−(Δx2)2−(Δx3)2→ds2=c2dt2−(dx1)2−(dx2)2−(dx3)2=0

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