reserve x,y for set;
reserve C,C9,D,D9,E for non empty set;
reserve c for Element of C;
reserve c9 for Element of C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve d9 for Element of D9;

theorem Th6:
  for X being set, x1,x2 being set holds <:X-->x1,X-->x2:> = X -->[x1,x2]
proof
  let X be set, x1,x2 be set;
  set f = X-->x1, g = X-->x2;
  set fg = <:f,g:>;
  now
    per cases;
    suppose
A1:   X = {};
      thus thesis by A1;
    end;
    suppose
A2:   X <> {};
      rng fg c= [:{x1},{x2}:];
      then
A3:   rng fg c= {[x1,x2]} by ZFMISC_1:29;
      set x = the Element of X;
A4:   dom f = X & dom g = X;
      then
A5:   dom fg = X by FUNCT_3:50;
      f.x = x1 & g.x = x2 by A2,FUNCOP_1:7;
      then fg.x = [x1,x2] by A2,A4,FUNCT_3:49;
      then [x1,x2] in rng fg by A2,A5,FUNCT_1:def 3;
      then {[x1,x2]} c= rng fg by ZFMISC_1:31;
      then rng fg = {[x1,x2]} by A3,XBOOLE_0:def 10;
      hence thesis by A5,FUNCOP_1:9;
    end;
  end;
  hence thesis;
end;
