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 st I = proj(1,1) qua Function"
 holds I is_differentiable_in x0 & diff(I,x0) = <*1*>
proof
  let I be Function of REAL,REAL-NS 1;
assume A1: I=proj(1,1) qua Function";
  I.jj = <*jj*> by A1,PDIFF_1:1;
then reconsider r = <*jj*> as Point of REAL-NS 1;
A2:for x be Real st x in ZR holds I/.x = x*r + 0.(REAL-NS 1)
   proof
    let x be Real;
     reconsider xx=x as Element of REAL by XREAL_0:def 1;
    assume x in ZR;
    I.jj in REAL 1 by A1,FUNCT_1:3,PDIFF_1:2;
    then
A3: <*1*> is Element of REAL 1 by A1,PDIFF_1:1;
    I/.xx = <*x*1*> by A1,PDIFF_1:1
        .= x*<*1*> by RVSUM_1:47;
    hence I/.x = x*r by A3,REAL_NS1:3
        .= x*r + 0.(REAL-NS 1) by RLVECT_1:4;
   end;
A4: ZR c= dom I by FUNCT_2:def 1;
   then
A5:I is_differentiable_on ZR
       & for x st x in ZR holds (I`|ZR).x = r by A2,NDIFF_3:21;
A6:  x0 in ZR by XREAL_0:def 1;
   hence I is_differentiable_in x0 by A5,NDIFF_3:10;
   r = (I`|ZR).x0 by A4,A2,A6,NDIFF_3:21
      .= diff(I,x0) by A5,A6,NDIFF_3:def 6;
   hence thesis;
end;
