
theorem LMN6:
for X be RealNormSpace, x be object holds
   x is additive homogeneous Function of X,REAL
 iff
   x is additive homogeneous Function of X,RNS_Real
proof
   let X be RealNormSpace, x be object;
   hereby assume A1: x is additive homogeneous Function of X,REAL; then
    reconsider f=x as linear-Functional of X;
    reconsider g=x as Function of X,RNS_Real by A1;
A2: for v,w be Element of X holds g.(v+w) = g.v + g.w
    proof
     let v,w be Element of X;
     thus g.(v+w) = f.v + f.w by HAHNBAN:def 2
                 .= g.v + g.w by BINOP_2:def 9;
    end;
    for v being VECTOR of X, r being Real holds g.(r*v) = r*g.v
    proof
     let v be VECTOR of X, r be Real;
     thus g.(r*v) = r*f.v by HAHNBAN:def 3
                 .= r*g.v by BINOP_2:def 11;
    end; then
    g is additive homogeneous by LOPBAN_1:def 5,A2;
    hence x is additive homogeneous Function of X,RNS_Real;
   end;
   assume B1: x is additive homogeneous Function of X,RNS_Real; then
   reconsider g=x as additive homogeneous Function of X,RNS_Real;
   reconsider f=x as Function of X,REAL by B1;
B2:for v,w be Element of X holds f.(v+w) = f.v + f.w
   proof
    let v,w be Element of X;
    thus f.(v+w) = g.v + g.w by VECTSP_1:def 20
                .= f.v + f.w by BINOP_2:def 9;
   end;
   for v being VECTOR of X, r being Real holds f.(r*v) = r*f.v
   proof
    let v be VECTOR of X, r be Real;
    thus f.(r*v) = r*g.v by LOPBAN_1:def 5
                .= r*f.v by BINOP_2:def 11;
   end;
   hence thesis by B2,HAHNBAN:def 2,def 3;
end;
