reserve x,x1,x2,y,y9,y1,y2,z,z1,z2 for object,P,X,X1,X2,Y,Y1,Y2,V,Z for set;

theorem Th25:
  for f being Function st x in dom f & x in X & f.x in Y holds <:f
  ,X,Y:>.x = f.x
proof
  let f be Function such that
A1: x in dom f and
A2: x in X and
A3: f.x in Y;
  x in dom(Y|`f) by A1,A3,FUNCT_1:54;
  then f.x = (Y|`f).x by FUNCT_1:55
    .= (Y|`f|X).x by A2,FUNCT_1:49;
  hence thesis;
end;
