
theorem LMN7:
for X be RealNormSpace, x be object holds
   x is Lipschitzian additive homogeneous Function of X,REAL
 iff
   x is Lipschitzian additive homogeneous Function of X,RNS_Real
proof
   let X be RealNormSpace, x be object;
   hereby
    assume A1: x is Lipschitzian additive homogeneous Function of X,REAL; then
    reconsider f=x as Lipschitzian linear-Functional of X;
    reconsider g=x as additive homogeneous Function of X,RNS_Real by A1,LMN6;
    consider K being Real such that
X1:   0 <= K &
      for v being VECTOR of X holds |. f.v .| <= K * ||. v .||
        by DUALSP01:def 9;
    for v being VECTOR of X holds ||. g.v .|| <= K * ||. v .||
    proof
     let v be VECTOR of X;
     |. f.v .| <= K * ||. v .|| by X1;
     hence ||. g.v .|| <= K * ||. v .|| by EUCLID:def 2;
    end;
    hence x is Lipschitzian additive homogeneous Function of X,RNS_Real
       by X1,LOPBAN_1:def 8;
   end;
   assume B1: x is Lipschitzian additive homogeneous
     Function of X,RNS_Real; then
   reconsider g=x as Lipschitzian additive homogeneous
     Function of X,RNS_Real;
   reconsider f=x as additive homogeneous Function of X,REAL by LMN6,B1;
   consider K being Real such that
X1: 0 <= K &
    for v being VECTOR of X holds ||. g.v .|| <= K * ||. v .||
      by LOPBAN_1:def 8;
   for v being VECTOR of X holds |. f.v .| <= K * ||. v .||
   proof
    let v be VECTOR of X;
    ||. g.v .|| <= K * ||. v .|| by X1;
    hence |. f.v .| <= K * ||. v .|| by EUCLID:def 2;
   end; then
   f is Lipschitzian additive homogeneous by X1;
   hence x is Lipschitzian additive homogeneous Function of X,REAL;
end;
