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 Th7:
  for f, g being Function st dom f = dom g holds pr1 <:f,g:> = f
proof
  let f, g be Function such that
A1: dom f = dom g;
A2: dom pr1 <:f,g:> = dom <:f,g:> by MCART_1:def 12;
A3: for x being object st x in dom pr1 <:f,g:> holds pr1 <:f,g:>.x = f.x
  proof
    let x be object such that
A4: x in dom pr1 <:f,g:>;
    thus pr1 <:f,g:>.x = (<:f,g:>.x)`1 by A2,A4,MCART_1:def 12
      .= [f.x,g.x]`1 by A2,A4,FUNCT_3:def 7
      .= f.x;
  end;
  dom <:f,g:> = dom f /\ dom g by FUNCT_3:def 7
    .= dom f by A1;
  hence thesis by A2,A3;
end;
