theorem
  (for a st a in X ex b st a [= b & b in Y) implies "\/"(X,C) [= "\/"(Y, C )
proof
  assume
A1: for a st a in X ex b st a [= b & b in Y;
  X is_less_than "\/"(Y,C)
  proof
    let a;
    assume a in X;
    then consider b such that
A2: a [= b and
A3: b in Y by A1;
    b [= "\/"(Y,C) by A3,Th38;
    hence thesis by A2,LATTICES:7;
  end;
  hence thesis by Def21;
end;
