
theorem Th3:
  for f, g being Function, X being set holds proj1 ([:f, g:].:X) c=
  f.:proj1 X & proj2 ([:f, g:].:X) c= g.:proj2 X
proof
  let f, g be Function, X be set;
A1: dom [:f, g:] = [:dom f, dom g:] by FUNCT_3:def 8;
  hereby
    let x be object;
    assume x in proj1 ([:f, g:].:X);
    then consider y being object such that
A2: [x, y] in [:f, g:].:X by XTUPLE_0:def 12;
    consider xy being object such that
A3: xy in dom [:f, g:] and
A4: xy in X and
A5: [x, y] = [:f, g:].xy by A2,FUNCT_1:def 6;
    consider x9,y9 being object such that
A6: x9 in dom f and
A7: y9 in dom g and
A8: xy = [x9,y9] by A1,A3,ZFMISC_1:def 2;
    [x, y] = [:f, g:].(x9,y9) by A5,A8
      .= [f.x9, g.y9] by A6,A7,FUNCT_3:def 8;
    then
A9: x = f.x9 by XTUPLE_0:1;
    x9 in proj1 X by A4,A8,XTUPLE_0:def 12;
    hence x in f.:proj1 X by A6,A9,FUNCT_1:def 6;
  end;
  let y be object;
  assume y in proj2 ([:f, g:].:X);
  then consider x being object such that
A10: [x, y] in [:f, g:].:X by XTUPLE_0:def 13;
  consider xy being object such that
A11: xy in dom [:f, g:] and
A12: xy in X and
A13: [x, y] = [:f, g:].xy by A10,FUNCT_1:def 6;
  consider x9,y9 being object such that
A14: x9 in dom f and
A15: y9 in dom g and
A16: xy = [x9,y9] by A1,A11,ZFMISC_1:def 2;
  [x, y] = [:f, g:].(x9,y9) by A13,A16
    .= [f.x9, g.y9] by A14,A15,FUNCT_3:def 8;
  then
A17: y = g.y9 by XTUPLE_0:1;
  y9 in proj2 X by A12,A16,XTUPLE_0:def 13;
  hence thesis by A15,A17,FUNCT_1:def 6;
end;
