reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem Th11:
diff(f,Z).0 = f|Z & diff(f,Z).1 = (f|Z)`| Z &
diff(f,Z).2 = ((f|Z)`| Z) `| Z
proof
   thus A1: diff(f,Z).0 = f|Z by Def5;
A2:diff_SP(0,S,T) = T by Th7; then
A3:modetrans(diff(f,Z).0,S,diff_SP(0,S,T)) = f|Z by A1,Def4;
   diff(f,Z).1 = diff(f,Z).( (0 qua Nat) + 1);
   hence A4: diff(f,Z).1 = (f|Z)`| Z by A3,A2,Def5;
A5:diff_SP(1,S,T) = R_NormSpace_of_BoundedLinearOperators(S,T)
     by Th7; then
A6:modetrans(diff(f,Z).1,S,diff_SP(1,S,T))
     = (f|Z)`| Z by A4,Def4;
   diff(f,Z).2 = diff(f,Z).( 1 + 1);
   hence diff(f,Z).2 = ((f|Z)`| Z) `| Z by A5,A6,Def5;
end;
