reserve i,j,k,n for Nat;
reserve x,x1,x2,x3,y1,y2,y3 for set;

theorem Th6:
  for f,g be one-to-one Function st dom f misses dom g & rng f
  misses rng g holds (f+*g)" = f" +* g"
proof
  let f,g be one-to-one Function such that
A1: dom f misses dom g and
A2: rng f misses rng g;
A3: (f+*g) is one-to-one by A2,FUNCT_4:92;
A4: for y,x being object
   holds y in rng (f+*g) & x = (f" +* g").y iff x in dom(f+*g) & y = (f+*g).x
  proof
    let y,x be object;
    thus y in rng (f+*g) & x = (f" +* g").y implies x in dom (f+*g) & y = (f+*
    g).x
    proof
A5:   rng (f+*g) c= rng f \/ rng g by FUNCT_4:17;
      assume that
A6:   y in rng (f+*g) and
A7:   x = (f" +* g").y;
      per cases by A6,A5,XBOOLE_0:def 3;
      suppose
A8:     y in rng f;
        then consider z being object such that
A9:     z in dom f and
A10:    y = f.z by FUNCT_1:def 3;
        dom (f +* g) = dom f \/ dom g by FUNCT_4:def 1;
        then
A11:    z in dom (f +* g) by A9,XBOOLE_0:def 3;
A12:    dom (f") = rng f & dom (g") = rng g by FUNCT_1:33;
        y = (f +* g).z & z = f".y by A1,A9,A10,FUNCT_1:32,FUNCT_4:16;
        hence thesis by A2,A7,A8,A11,A12,FUNCT_4:16;
      end;
      suppose
A13:    y in rng g;
A14:    dom (f") = rng f & dom (g") = rng g by FUNCT_1:33;
        consider z being object such that
A15:    z in dom g and
A16:    y = g.z by A13,FUNCT_1:def 3;
        z = g".y by A15,A16,FUNCT_1:32;
        then z = (g" +* f").y by A2,A13,A14,FUNCT_4:16;
        then
A17:    z = x by A2,A7,A14,FUNCT_4:35;
        dom (f +* g) = dom f \/ dom g & y = (g +* f).z by A1,A15,A16,FUNCT_4:16
,def 1;
        hence thesis by A1,A15,A17,FUNCT_4:35,XBOOLE_0:def 3;
      end;
    end;
    thus x in dom (f+*g) & y = (f+*g).x implies y in rng (f+*g) & x = (f" +* g
    ").y
    proof
A18:  dom (f+*g) = dom f \/ dom g by FUNCT_4:def 1;
      assume that
A19:  x in dom (f+*g) and
A20:  y = (f+*g).x;
      per cases by A19,A18,XBOOLE_0:def 3;
      suppose
A21:    x in dom f;
        then not x in dom g by A1,XBOOLE_0:3;
        then
A22:    y = f.x by A20,FUNCT_4:11;
        then
A23:    x = f".y by A21,FUNCT_1:32;
A24:    dom (f") = rng f & dom (g") = rng g by FUNCT_1:33;
A25:    y in rng f by A21,A22,FUNCT_1:def 3;
        then y in rng f \/ rng g by XBOOLE_0:def 3;
        hence thesis by A1,A2,A25,A23,A24,FRECHET:35,FUNCT_4:16,PARTFUN1:56;
      end;
      suppose
A26:    x in dom g;
        then
A27:    y = g.x by A20,FUNCT_4:13;
        then
A28:    y in rng g by A26,FUNCT_1:def 3;
        then
A29:    y in rng f \/ rng g by XBOOLE_0:def 3;
A30:    dom (f") = rng f & dom (g") = rng g by FUNCT_1:33;
        x = g".y by A26,A27,FUNCT_1:32;
        then x = (g" +* f").y by A2,A28,A30,FUNCT_4:16;
        hence thesis by A1,A2,A29,A30,FRECHET:35,FUNCT_4:35,PARTFUN1:56;
      end;
    end;
  end;
  dom (f" +* g") = dom (f") \/ dom (g") by FUNCT_4:def 1
    .= rng f \/ dom (g") by FUNCT_1:33
    .= rng f \/ rng g by FUNCT_1:33
    .= rng (f +* g) by A1,FRECHET:35,PARTFUN1:56;
  hence thesis by A3,A4,FUNCT_1:32;
end;
