
theorem Th1:
  for X being set, S being Subset of id X holds proj1 S = proj2 S
proof
  let X be set, S be Subset of id X;
  now
    let x be object;
    hereby
      assume x in proj1 S;
      then consider y being object such that
A1:   [x, y] in S by XTUPLE_0:def 12;
      x = y by A1,RELAT_1:def 10;
      hence x in proj2 S by A1,XTUPLE_0:def 13;
    end;
    assume x in proj2 S;
    then consider y being object such that
A2: [y, x] in S by XTUPLE_0:def 13;
    x = y by A2,RELAT_1:def 10;
    hence x in proj1 S by A2,XTUPLE_0:def 12;
  end;
  hence thesis by TARSKI:2;
end;
