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
  x,y,w,z are_mutually_distinct implies
  rng ((x,y,w,z) --> (a,b,c,d)) = {a,b,c,d}
proof
  set h=(x,y,w,z) --> (a,b,c,d);
  assume
A1: x,y,w,z are_mutually_distinct;
A2: rng h c= {a,b,c,d} by Th138;
  {a,b,c,d} c= rng h
proof
  set h=(x,y,w,z) --> (a,b,c,d);
  let y1 be object;
  assume y1 in {a,b,c,d}; then
A3: y1=a or y1=b or y1=c or y1=d by ENUMSET1:def 2;
A4: dom h = {x,y,w,z} by Th137;
A5: h.x=y1 or h.y=y1 or h.w=y1 or h.z=y1 by A1,A3,Th139,Th140,Th141,Th142;
A6: x in dom h by A4,ENUMSET1:def 2;
A7: y in dom h by A4,ENUMSET1:def 2;
A8: w in dom h by A4,ENUMSET1:def 2;
  z in dom h by A4,ENUMSET1:def 2;
  hence thesis by A5,A6,A7,A8,FUNCT_1:def 3;
end;
  hence thesis by A2;
end;
