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 Th33:
for I be Function of REAL,REAL-NS 1 st I=proj(1,1) qua Function"
 holds I is_continuous_in x0
proof
 let I be Function of REAL,REAL-NS 1;
 assume A1: I=proj(1,1) qua Function";
A2: I is Function of REAL,REAL 1 by REAL_NS1:def 4;
A3: dom I = REAL by FUNCT_2:def 1;
   then
A4: x0 in dom I by XREAL_0:def 1;
 reconsider y0=x0 as Element of REAL by XREAL_0:def 1;
 now let r be Real;
  assume A5: 0 < r;
  reconsider s1=r as Real;
  take s=s1;
  thus 0 < s by A5;
  thus for y1 be Real st y1
  in dom I & |.y1-y0.|<s holds ||. I/.y1 - I/.y0 .||<r
  proof
    let y1 be Real;
    assume A6: y1 in dom I & |.y1- y0.|<s;
    reconsider x1=y1 as Element of REAL by XREAL_0:def 1;
A7:  I.x1 = I/.y1 & I.x0 = I/.x0 by A4,A6,PARTFUN1:def 6;
    then reconsider Ia = I.x1, Ib = I.x0 as VECTOR of REAL-NS 1;
    Ia-Ib = I.(x1-x0) by A1,A2,PDIFF_1:3;
    hence thesis by A6,A1,A2,A7,PDIFF_1:3;
  end;
 end;
 hence thesis by A3,NFCONT_3:8;
end;
