theorem
  f = h & x = z implies (f is_partial_differentiable_in x,i,j iff
  h is_partial_differentiable_in z,i,j)
proof
  assume that
A1: f=h and
A2: x=z;
A3: Proj(i,m).x=<*proj(i,m).z*> by A2,Def4;
  Proj(j,n)*f*reproj(i,x) = <>*(proj(j,n)*h*reproj(i,z)) by A1,A2,Th22;
  hence thesis by A3,Th7,Th8;
end;
