reserve Y for RealNormSpace;

theorem NDIFF435:
for Y be RealNormSpace, 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_continuous_in x0 iff g is_continuous_in y0
proof
   let Y be RealNormSpace, 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;
   thus f is_continuous_in x0 implies g is_continuous_in y0
   proof
    reconsider I= proj(1,1) qua Function" as Function of REAL,REAL-NS 1
      by PDIFF_1:2,REAL_NS1:def 4;
A2: J*I = id REAL by A1,Lm2,FUNCT_1:39;
    f*I = g*(id REAL) by A1,A2,RELAT_1:36; then
A3: f*I = g by FUNCT_2:17;
    I/.y0 = x0 by A1,PDIFF_1:1;
    hence thesis by A3,NDIFF_4:33,A1,NFCONT_3:15;
   end;
   J/.x0 = y0 by A1,PDIFF_1:1;
   hence thesis by A1,NDIFF_4:32,NDIFF_4:34;
end;
