reserve i,n,m for Nat;

theorem Th12:
for f be Function of REAL m,REAL n, g be Function of REAL-NS m,REAL-NS n st
 f=g holds f is additive iff g is additive
proof
   let f be Function of REAL m,REAL n,
       g be Function of REAL-NS m,REAL-NS n;
   assume A1: f=g;
   hereby assume A2:f is additive;
    now let x,y be Point of REAL-NS m;
     reconsider x1=x,y1=y as Element of REAL m by REAL_NS1:def 4;
     g.(x+y) = f.(x1+y1) by A1,REAL_NS1:2
            .= f.x1+ f.y1 by A2;
     hence g.(x+y) = g.x+ g.y by A1,REAL_NS1:2;
    end;
    hence g is additive by VECTSP_1:def 20;
   end;
   assume A3:g is additive;
   now let x,y be Element of REAL m;
    reconsider x1=x, y1=y as Point of REAL-NS m by REAL_NS1:def 4;
    f.(x+y) = g.(x1+y1) by A1,REAL_NS1:2
           .= g.x1+ g.y1 by A3,VECTSP_1:def 20;
    hence f.(x+y) = f.x+ f.y by A1,REAL_NS1:2;
   end;
   hence f is additive;
end;
