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 Th30:
  f1 is_partial_differentiable_in x,i & f2
is_partial_differentiable_in x,i implies f1-f2 is_partial_differentiable_in x,i
  & partdiff(f1-f2,x,i)=partdiff(f1,x,i)-partdiff(f2,x,i)
proof
  assume that
A1: f1 is_partial_differentiable_in x,i and
A2: f2 is_partial_differentiable_in x,i;
A3: f1*reproj(i,x) is_differentiable_in Proj(i,m).x by A1;
A4: f2*reproj(i,x) is_differentiable_in Proj(i,m).x by A2;
  (f1-f2)*reproj(i,x) = f1*reproj(i,x)-f2*reproj(i,x) by Th26;
  then (f1-f2)*reproj(i,x) is_differentiable_in Proj(i,m).x by A3,A4,NDIFF_1:36
;
  hence f1-f2 is_partial_differentiable_in x,i;
  diff(f1*reproj(i,x)-f2*reproj(i,x),Proj(i,m).x) = partdiff(f1-f2,x,i) by Th26
;
  hence thesis by A3,A4,NDIFF_1:36;
end;
