reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve y,X for set;
reserve x,x0,x1,x2,g,g1,g2,r,r1,s,p,p1,p2 for Real;
reserve i,m,k for Element of NAT;
reserve n,k for non zero Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq,seq1 for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve g,g1,g2 for PartFunc of REAL,REAL-NS n;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;
reserve GR,R for RestFunc of REAL-NS n;
reserve DFG,L for LinearFunc of REAL-NS n;

theorem
for 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,REAL-NS n,
      f be PartFunc of REAL-NS 1,REAL-NS n
   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 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,REAL-NS n,
      f be PartFunc of REAL-NS 1,REAL-NS n;
  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,Th33,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,Th32,A1,Th34;
end;
