
theorem Th6:
  for p, q being Function st q is disjoint_valued & p c= q holds
    p is disjoint_valued
proof
  let p, q be Function;
  assume that
A1: q is disjoint_valued and
A2: p c= q;
  for x, y being object st x <> y holds p.x misses p.y
  proof
    let x, y be object;
    assume
A3: x <> y;
    assume
A4: p.x meets p.y;
    x in dom p & y in dom p
    proof
      assume not x in dom p or not y in dom p;
      then p.x = {} or p.y = {} by FUNCT_1:def 2;
      hence thesis by A4;
    end;
    then p.x = q.x & p.y = q.y by A2,GRFUNC_1:2;
    hence thesis by A1,A3,A4;
  end;
  hence thesis;
end;
