theorem
  for a,b,c,d,x,y,z,w,x9,y9,z9,w9 being object
   st a,b,c,d are_mutually_distinct &
  (a,b,c,d) --> (x,y,z,w) = (a,b,c,d) --> (x9,y9,z9,w9)
  holds x = x9 & y = y9 & z=z9 & w=w9
proof
  let a,b,c,d,x,y,z,w,x9,y9,z9,w9 be object such that
A1: a,b,c,d are_mutually_distinct and
A2: (a,b,c,d) --> (x,y,z,w) = (a,b,c,d) --> (x9,y9,z9,w9);
A3: x=((a,b,c,d) --> (x,y,z,w)).a by A1,Th142
    .=x9 by A1,A2,Th142;
A4: y=((a,b,c,d) --> (x,y,z,w)).b by A1,Th141
    .=y9 by A1,A2,Th141;
A5: z=((a,b,c,d) --> (x,y,z,w)).c by A1,Th140
    .=z9 by A1,A2,Th140;
  w=((a,b,c,d) --> (x,y,z,w)).d by Th139
    .=w9 by A2,Th139;
  hence thesis by A3,A4,A5;
end;
