reserve i, j, k, c, m, n for Nat,
  a, x, y, z, X, Y for set,
  D, E for non empty set,
  R for Relation,
  f, g for Function,
  p, q for FinSequence;

theorem Th10:
  X is functional compatible iff union X is Function
proof
  now
    assume
A1: union X is Function;
    thus X is functional
    proof
      let f be object;
      assume
A2:   f in X;
      reconsider f as set by TARSKI:1;
A3:   f is Function-like
      proof
        let x, y1, y2 be object;
        assume that
A4:     [x,y1] in f and
A5:     [x, y2] in f;
A6:     [x,y2] in union X by A2,A5,TARSKI:def 4;
        [x,y1] in union X by A2,A4,TARSKI:def 4;
        hence thesis by A1,A6,FUNCT_1:def 1;
      end;
      f is Relation-like
      proof
        let x be object;
        assume x in f;
        then x in union X by A2,TARSKI:def 4;
        hence thesis by A1,RELAT_1:def 1;
      end;
      hence thesis by A3;
    end;
    thus X is compatible
    proof
      let f,g be Function such that
A7:   f in X and
A8:   g in X;
      let x be object;
      assume
A9:   x in dom f /\ dom g;
      then
A10:  x in dom g by XBOOLE_0:def 4;
      then consider y2 being object such that
A11:  [x,y2] in g by XTUPLE_0:def 12;
A12:  [x,y2] in union X by A8,A11,TARSKI:def 4;
A13:  x in dom f by A9,XBOOLE_0:def 4;
      then consider y1 being object such that
A14:  [x,y1] in f by XTUPLE_0:def 12;
A15:   y1 is set by TARSKI:1;
      [x,y1] in union X by A7,A14,TARSKI:def 4;
      then
A16:  y1 = y2 by A1,A12,FUNCT_1:def 1;
      thus f.x = y1 by A13,A14,FUNCT_1:def 2,A15
        .= g.x by A10,A11,A16,FUNCT_1:def 2,A15;
    end;
  end;
  hence thesis;
end;
