reserve Y for RealNormSpace;

theorem
for Y be RealNormSpace, I be Function of REAL,REAL-NS 1,
    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 I=proj(1,1) qua Function"
    & x0 in dom f & y0 in dom g & x0=<*y0*> & f*I = g holds
  f is_continuous_in x0 iff g is_continuous_in y0
proof
   let Y be RealNormSpace, I be Function of REAL,REAL-NS 1,
       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: I=proj(1,1) qua Function"
        & x0 in dom f & y0 in dom g & x0=<*y0*> & f*I = g;
   reconsider J= proj(1,1) as Function of REAL-NS 1,REAL by Lm1;
   thus f is_continuous_in x0 implies g is_continuous_in y0
   proof
    I/.y0 = x0 by A1,PDIFF_1:1;
    hence thesis by A1,NDIFF_4:33,NFCONT_3:15;
   end;
A2: I*J = id REAL-NS 1 by A1,Lm2,Lm1,FUNCT_1:39;
A3: g*J = f*(id REAL-NS 1) by A2,A1,RELAT_1:36
       .= f by FUNCT_2:17;
    J/.x0 = y0 by A1,PDIFF_1:1;
    hence thesis by A3,NDIFF_4:32,A1,NDIFF_4:34;
end;
