reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;
reserve A,B for set;
reserve x,y,i,j,k for object;
reserve x for set;
reserve x for object;
reserve A1,A2,B1,B2 for non empty set,
  f for Function of A1,B1,
  g for Function of A2,B2,
  Y1 for non empty Subset of A1,
  Y2 for non empty Subset of A2;
reserve a,b,c,x,y,z,w,d for object;

theorem
  for a1,a2,a3,b1,b2,b3 being object
    st a1,a2,a3 are_mutually_distinct
   holds
  rng ((a1,a2,a3)-->(b1,b2,b3)) = {b1,b2,b3}
  proof
   let a1,a2,a3,b1,b2,b3 be object;
   assume
A1:  a1,a2,a3 are_mutually_distinct;
    set f = (a1,a2,a3)-->(b1,b2,b3);
    thus rng (f) c= {b1,b2,b3}
    proof
      let x be object; assume x in rng f; then
      consider y being object such that
A2:   y in dom f & x = f.y by FUNCT_1:def 3;
      dom f = {a1,a2,a3} by Th128; then
      y = a1 or y = a2 or y = a3 by A2,ENUMSET1:def 1; then
      x = b1 or x = b2 or x = b3 by A2,A1,Th134,Th135;
      hence thesis by ENUMSET1:def 1;
    end;
    let x be object; assume x in {b1,b2,b3}; then
A3: x = b1 or x = b2 or x = b3 by ENUMSET1:def 1;
A4: a1 in {a1,a2,a3} & a2 in {a1,a2,a3} & a3 in {a1,a2,a3} by ENUMSET1:def 1;
A5: dom f = {a1,a2,a3} by Th128;
    f.a1 = b1 & f.a2 = b2 & f.a3 = b3 by A1,Th134,Th135;
    hence thesis by A3,A4,A5,FUNCT_1:def 3;
  end;
