
theorem Th78:
  for x1,y1, x2,y2 being set holds x1 U+ y1 c= x2 U+ y2 iff x1 c= x2 & y1 c= y2
proof
  let x1,y1, x2,y2 be set;
  hereby
    assume
A1: x1 U+ y1 c= x2 U+ y2;
    thus x1 c= x2
    proof
      let a be object;
      assume a in x1;
      then [a,1] in x1 U+ y1 by Th76;
      hence thesis by A1,Th76;
    end;
    thus y1 c= y2
    proof
      let a be object;
      assume a in y1;
      then [a,2] in x1 U+ y1 by Th77;
      hence thesis by A1,Th77;
    end;
  end;
  assume
A2: x1 c= x2 & y1 c= y2;
  let a be object;
  assume
A3: a in x1 U+ y1;
  then
A4: a`2 = 1 & a`1 in x1 or a`2 = 2 & a`1 in y1 by Th75;
  a = [a`1,a`2] by A3,Th75;
  hence thesis by A2,A4,Th76,Th77;
end;
