reserve Y for RealNormSpace;

theorem FTh44:
for J be Function of REAL-NS 1,REAL,
      x0 be Point of REAL-NS 1, y0 be Element of REAL,
      g be PartFunc of REAL,Y, f be PartFunc of REAL-NS 1,Y
   st J=proj(1,1) & x0 in dom f & y0 in dom g & x0=<*y0*> & f = g*J holds
  f is_differentiable_in x0 iff g is_differentiable_in y0
proof
   let J be Function of REAL-NS 1,REAL,
       x0 be Point of REAL-NS 1, y0 be Element of REAL,
       g be PartFunc of REAL,Y, f be PartFunc of REAL-NS 1,Y;
   assume A1: J=proj(1,1) & x0 in dom f & y0 in dom g & x0=<*y0*> & f = g*J;
   reconsider I= (proj(1,1) qua Function") as Function of REAL,REAL-NS 1
               by PDIFF_1:2,REAL_NS1:def 4;
   J*I = id REAL by A1,Lm2,FUNCT_1:39; then
   f*I = g*(id REAL) by A1,RELAT_1:36
      .= g by FUNCT_2:17;
   hence thesis by A1,FTh43;
end;
