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;

theorem
  for g1 be PartFunc of REAL n,REAL 1 st g1 = <>*g & g1
  is_partial_differentiable_in y,i holds partdiff(g1,y,i) = <*partdiff(g,y,i)*>
proof
  let g1 be PartFunc of REAL n,REAL 1;
  assume that
A1: g1=<>*g and
A2: g1 is_partial_differentiable_in y,i;
  reconsider y9=y as Point of REAL-NS n by REAL_NS1:def 4;
  the carrier of REAL-NS 1 = REAL 1 by REAL_NS1:def 4;
  then reconsider h=g1 as PartFunc of REAL-NS n,REAL-NS 1 by REAL_NS1:def 4;
A3: h is_partial_differentiable_in y9,i by A2;
  then partdiff(h,y9,i).<*1*>= partdiff(g1,y,i) by Th17;
  hence thesis by A1,A3,Th15;
end;
