reserve S, T, Y for non empty TopSpace,
  s, s1, s2, s3 for Point of S,
  t, t1, t2, t3 for Point of T,
  l1, l2 for Path of [s1,t1],[s2,t2],
  H for Homotopy of l1 ,l2;

theorem Th8:
  for f, g being Function st dom f = dom g holds pr2 <:f,g:> = g
proof
  let f, g be Function such that
A1: dom f = dom g;
A2: dom pr2 <:f,g:> = dom <:f,g:> by MCART_1:def 13;
A3: for x being object st x in dom pr2 <:f,g:> holds pr2 <:f,g:>.x = g.x
  proof
    let x be object such that
A4: x in dom pr2 <:f,g:>;
    thus pr2 <:f,g:>.x = (<:f,g:>.x)`2 by A2,A4,MCART_1:def 13
      .= [f.x,g.x]`2 by A2,A4,FUNCT_3:def 7
      .= g.x;
  end;
  dom <:f,g:> = dom f /\ dom g by FUNCT_3:def 7
    .= dom g by A1;
  hence thesis by A2,A3;
end;
