reserve n for Nat,
        p,p1,p2 for Point of TOP-REAL n,
        x for Real;
reserve n,m for non zero Nat;
reserve i,j for Nat;
reserve f for PartFunc of REAL-NS m,REAL-NS n;
reserve g for PartFunc of REAL m,REAL n;
reserve h for PartFunc of REAL m,REAL;
reserve x for Point of REAL-NS m;
reserve y for Element of REAL m;
reserve X for set;

theorem Th22:
g is_differentiable_in y & 1 <=i & i <= m implies
g is_partial_differentiable_in y,i
& partdiff(g,y,i) = (diff(g,y)*reproj(i,0.(REAL-NS m))).<*1*>
proof
   assume A1: g is_differentiable_in y & 1 <=i & i <= m;
   then consider f be PartFunc of REAL-NS m,REAL-NS n,
            x be Point of REAL-NS m such that
A2: f=g & x=y & f is_differentiable_in x;
A3:ex f2 be PartFunc of REAL-NS m,REAL-NS n, x2 be Point of REAL-NS m st
   f2=g & x2=y & diff(g,y) = diff(f2,x2) by A1,PDIFF_1:def 8;
A4:f is_partial_differentiable_in x,i &
   partdiff(f,x,i) = diff(f,x)*reproj(i,0.(REAL-NS m)) by Th21,A2,A1;
   then g is_partial_differentiable_in y,i by A2;
   then
 ex f1 be PartFunc of REAL-NS m,REAL-NS n, x1 be Point of REAL-NS m st
    f1=g & x1=y & partdiff(g,y,i) = partdiff(f1,x1,i).<*1*>
      by PDIFF_1:def 14;
   hence thesis by A4,A3,A2;
end;
