reserve X,Y,Z,X1,X2,Y1,Y2 for set, x,y,z,t,x1,x2 for object,
  f,g,h,f1,f2,g1,g2 for Function;

theorem Th17:
  [:X,Y:] <> {} & dom f = [:X,Y:] implies dom curry f = X & dom curry' f = Y
proof
  assume that
A1: [:X,Y:] <> {} and
A2: dom f = [:X,Y:];
  dom curry f = proj1 dom f by Def1;
  hence dom curry f = X by A1,A2,Th3;
  thus dom curry' f = proj1 dom ~f by Def1
    .= proj1 [:Y,X:] by A2,FUNCT_4:46
    .= Y by A1,Th3;
end;
