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;
