theorem
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,REAL-NS n,
      f be PartFunc of REAL-NS 1,REAL-NS n
   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 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,REAL-NS n,
      f be PartFunc of REAL-NS 1,REAL-NS n;
  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;
A3: f*I = g*(id REAL) by A1,A2,RELAT_1:36
       .= g by FUNCT_2:17;
    I/.y0 = x0 by A1,PDIFF_1:1;
    hence thesis by A3,Th33,A1,NFCONT_3:15;
  end;
  J/.x0 = y0 by A1,PDIFF_1:1;
  hence thesis by A1,Th32,Th34;
end;
