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 Th13:
 for x,y being object holds
  [x,y] in dom f & g = (curry f).x implies y in dom g & g.y = f.(x ,y)
proof let x,y be object;
  assume that
A1: [x,y] in dom f and
A2: g = (curry f).x;
  x in proj1 dom f by A1,XTUPLE_0:def 12;
  then
A3: ex h st (curry f).x = h & dom h = proj2 (dom f /\ [:{x},proj2 dom f :])
  & for y st y in dom h holds h.y = f.(x,y) by Def1;
  y in proj2 dom f by A1,XTUPLE_0:def 13;
  then [x,y] in [:{x},proj2 dom f:] by ZFMISC_1:105;
  then [x,y] in dom f /\ [:{x},proj2 dom f:] by A1,XBOOLE_0:def 4;
  hence y in dom g by A2,A3,XTUPLE_0:def 13;
  hence thesis by A2,A3;
end;
