reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve y,X for set;
reserve x,x0,x1,x2,g,g1,g2,r,r1,s,p,p1,p2 for Real;
reserve i,m,k for Element of NAT;
reserve n,k for non zero Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq,seq1 for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve g,g1,g2 for PartFunc of REAL,REAL-NS n;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;

theorem
  Z c= dom (f1-f2) & f1 is_differentiable_on Z & f2 is_differentiable_on Z
  implies f1-f2 is_differentiable_on Z &
  for x st x in Z holds ((f1-f2)`|Z).x = diff(f1,x) - diff(f2,x)
proof
A1: f1-f2 = f1+-f2 by Th1;
    assume that
A2: Z c= dom (f1-f2) and
A3: f1 is_differentiable_on Z and
A4: f2 is_differentiable_on Z;
A5: -f2 is_differentiable_on Z by A4,Th14;
    hence f1-f2 is_differentiable_on Z by A1,A2,A3,Th15;
    let x;
    assume
A6: x in Z;
    then
A7: f2 is_differentiable_in x by A4,Th5;
    thus ((f1-f2)`|Z).x = diff(f1,x) + diff(-f2,x) by A1,A2,A3,A5,A6,Th15
    .= diff(f1,x) - diff(f2,x) by A7,Th10;
end;
