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 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;
