reserve i,n,m for Nat;

theorem Th13:
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 homogeneous iff g is homogeneous
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 homogeneous;
    now let x be Point of REAL-NS m ,r be Real;
     reconsider x1=x as Element of REAL m by REAL_NS1:def 4;
     g.(r*x) = f.(r*x1) by A1,REAL_NS1:3
            .= r*(f.x1) by A2;
     hence g.(r*x) = r*(g.x) by A1,REAL_NS1:3;
    end;
    hence g is homogeneous;
   end;
   assume A3:g is homogeneous;
   now let x being Element of REAL m, r be Real;
    reconsider x1=x as Point of REAL-NS m by REAL_NS1:def 4;
    f.(r*x) = g.(r*x1) by A1,REAL_NS1:3
           .= r*(g.x1) by A3;
    hence f.(r*x) = r*(f.x) by A1,REAL_NS1:3;
   end;
   hence f is homogeneous;
end;
