reserve f for PartFunc of REAL-NS 1,REAL-NS 1;
reserve g for PartFunc of REAL,REAL;
reserve x for Point of REAL-NS 1;
reserve y for Real;
reserve m,n for non zero Nat;
reserve i,j for Nat;
reserve f for PartFunc of REAL-NS n,REAL-NS 1;
reserve g for PartFunc of REAL n,REAL;
reserve x for Point of REAL-NS n;
reserve y for Element of REAL n;
reserve X for set;
reserve r for Real;
reserve f,f1,f2 for PartFunc of REAL-NS m,REAL-NS n;
reserve g,g1,g2 for PartFunc of REAL n,REAL;
reserve h for PartFunc of REAL m,REAL n;
reserve x for Point of REAL-NS m;
reserve y for Element of REAL n;
reserve z for Element of REAL m;

theorem Th32:
  f is_partial_differentiable_in x,i implies r(#)f
  is_partial_differentiable_in x,i & partdiff((r(#)f),x,i) = r*partdiff(f,x,i)
proof
  assume f is_partial_differentiable_in x,i;
  then
A1: f*reproj(i,x) is_differentiable_in Proj(i,m).x;
  r(#)(f*reproj(i,x)) = (r(#)f)*reproj(i,x) by Th27;
  then (r(#)f)*reproj(i,x) is_differentiable_in Proj(i,m).x by A1,NDIFF_1:37;
  hence r(#)f is_partial_differentiable_in x,i;
  diff(r(#)(f*reproj(i,x)),Proj(i,m).x) = partdiff(r(#)f,x,i) by Th27;
  hence thesis by A1,NDIFF_1:37;
end;
