reserve A, B, X, Y for set;
reserve R, S, T for non empty TopSpace;

theorem Th37:
  id the carrier of T = {p where p is Point of [:T,T:]: pr1(the
  carrier of T,the carrier of T).p = pr2(the carrier of T,the carrier of T).p}
proof
  set f = pr1(the carrier of T,the carrier of T), g = pr2(the carrier of T,the
  carrier of T);
A1: the carrier of [:T,T:] = [:the carrier of T,the carrier of T:] by
BORSUK_1:def 2;
  hereby
    let a be object;
    assume
A2: a in id the carrier of T;
    then consider x, y being object such that
A3: x in the carrier of T and
A4: y in the carrier of T and
A5: a = [x,y] by ZFMISC_1:def 2;
A6: x = y by A2,A5,RELAT_1:def 10;
    f.(x,y) = x by A3,A4,FUNCT_3:def 4
      .= g.(x,y) by A3,A6,FUNCT_3:def 5;
    hence a in {p where p is Point of [:T,T:]: f.p = g.p} by A1,A2,A5;
  end;
  let a be object;
  assume a in {p where p is Point of [:T,T:]: f.p = g.p};
  then consider p being Point of [:T,T:] such that
A7: a = p and
A8: f.p = g.p;
  consider x, y being object such that
A9: x in the carrier of T and
A10: y in the carrier of T and
A11: p = [x,y] by A1,ZFMISC_1:def 2;
A12: f.(x,y) = g.(x,y) by A8,A11;
  x = f.(x,y) by A9,A10,FUNCT_3:def 4
    .= y by A9,A10,A12,FUNCT_3:def 5;
  hence thesis by A7,A9,A11,RELAT_1:def 10;
end;
