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
  f=h & x=z & f is_partial_differentiable_in x,i,j implies
  partdiff(f,x,i,j).<*1*> = <*partdiff(h,z,i,j)*>
proof
  assume that
A1: f=h and
A2: x=z and
A3: f is_partial_differentiable_in x,i,j;
A4: Proj(i,m).x=<*proj(i,m).z*> by A2,Def4;
A5: Proj(j,n)*f*reproj(i,x) is_differentiable_in Proj(i,m).x by A3;
  Proj(j,n)*f*reproj(i,x) =<>*(proj(j,n)*h*reproj(i,z)) by A1,A2,Th22;
  hence thesis by A4,A5,Th10;
end;
