theorem Th45:
  X c= Y implies "\/"(X,C) [= "\/"(Y,C) & "/\"(Y,C) [= "/\"(X,C)
proof
  assume
A1: X c= Y;
  X is_less_than "\/"(Y,C)
  proof
    let a;
    assume
A2: a in X;
    Y is_less_than "\/"(Y,C) by Def21;
    hence thesis by A1,A2;
  end;
  hence "\/"(X,C) [= "\/"(Y,C) by Def21;
  "/\"(Y,C) is_less_than X
  proof
    let a;
    assume
A3: a in X;
    "/\"(Y,C) is_less_than Y by Th34;
    hence thesis by A1,A3;
  end;
  hence thesis by Th34;
end;
