
theorem Th19:
  for G1, G2 being _Graph holds G1 tolerates G2 iff {G1,G2} is \/-tolerating
proof
  let G1, G2 be _Graph;
  now
    assume A1: G1 tolerates G2;
    let G3, G4 be Element of {G1,G2};
    per cases by TARSKI:def 2;
    suppose G3 = G1 & G4 = G1;
      hence G3 tolerates G4;
    end;
    suppose G3 = G1 & G4 = G2;
      hence G3 tolerates G4 by A1;
    end;
    suppose G3 = G2 & G4 = G1;
      hence G3 tolerates G4 by A1;
    end;
    suppose G3 = G2 & G4 = G2;
      hence G3 tolerates G4;
    end;
  end;
  hence G1 tolerates G2 implies {G1,G2} is \/-tolerating;
  assume A2: {G1,G2} is \/-tolerating;
  G1 in {G1,G2} & G2 in {G1,G2} by TARSKI:def 2;
  hence G1 tolerates G2 by A2;
end;
