reserve x, X, Y for set;
reserve L for complete LATTICE,
  a for Element of L;

theorem Th24: ::Proposition 2.2 (i)  (variant 1)   cf YELLOW_0
  for L being non empty RelStr holds (for X holds ex_sup_of X,L)
  implies L is complete
proof
  let L be non empty RelStr;
  assume
A1: for X holds ex_sup_of X,L;
  L is complete
  proof
    let X be set;
    take a = "\/"(X, L);
A2: ex_sup_of X,L by A1;
    hence X is_<=_than a by YELLOW_0:def 9;
    thus thesis by A2,YELLOW_0:def 9;
  end;
  hence thesis;
end;
