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 Th32:
for J be Function of REAL-NS 1,REAL,
      x0 be Point of REAL-NS 1 st J=proj(1,1)
 holds J is_continuous_in x0
proof
  let J be Function of REAL-NS 1,REAL,
     x0 be Point of REAL-NS 1;
  assume A1: J=proj(1,1);
A2: dom J =the carrier of REAL-NS 1 by FUNCT_2:def 1;
  now let r be Real;
    assume A3: 0 < r;
    take s=r;
    thus 0 < s by A3;
    thus for x1 be Point of REAL-NS 1 st x1
        in dom J & ||. x1- x0 .|| <s holds |.J/.x1-J/.x0.|<r
    proof
      let x1 be Point of REAL-NS 1;
      J/.x1-J/.x0 = J.(x1-x0) by A1,PDIFF_1:4;
      hence thesis by A1,PDIFF_1:4;
   end;
 end;
 hence thesis by A2,NFCONT_1:8;
end;
