
theorem Th1:
  for X being functional set st for f, g being Function st f in X &
  g in X holds f tolerates g holds union X is Function
proof
  let X be functional set;
  assume
A1: for f, g being Function st f in X & g in X holds f tolerates g;
A2: union X is Function-like
  proof
    let x,y1,y2 be object;
    assume that
A3: [x,y1] in union X and
A4: [x,y2] in union X and
A5: y1<>y2;
    consider gx being set such that
A6: [x, y2] in gx and
A7: gx in X by A4,TARSKI:def 4;
    consider fx being set such that
A8: [x, y1] in fx and
A9: fx in X by A3,TARSKI:def 4;
    reconsider fx, gx as Function by A9,A7;
    fx tolerates gx by A1,A9,A7;
    then ex h being Function st fx c= h & gx c= h by PARTFUN1:52;
    hence contradiction by A5,A8,A6,FUNCT_1:def 1;
  end;
  union X is Relation-like
  proof
    let x be object;
    assume x in union X;
    then ex ux being set st x in ux & ux in X by TARSKI:def 4;
    hence thesis by RELAT_1:def 1;
  end;
  hence thesis by A2;
end;
