
theorem NISOM08:
for X,Y be RealNormSpace,
    L be Lipschitzian LinearOperator of X,Y,
    y be Lipschitzian linear-Functional of Y
 holds y*L is Lipschitzian linear-Functional of X
proof
   let X,Y be RealNormSpace,
       L be Lipschitzian LinearOperator of X,Y,
       y be Lipschitzian linear-Functional of Y;
   consider M being Real such that
AS1: 0 <= M &
     for x being VECTOR of X holds
      ||. L.x .|| <= M * ||. x .|| by LOPBAN_1:def 8;
D1:dom L = the carrier of X by FUNCT_2:def 1;
   set x = y * L;
P6:for v,w be VECTOR of X holds x.(v+w) = x.v + x.w
   proof
    let v,w be VECTOR of X;
    thus x.(v+w) = y.(L.(v+w)) by D1,FUNCT_1:13
                .= y.(L.v+L.w) by VECTSP_1:def 20
                .= y.(L.v) + y.(L.w) by HAHNBAN:def 2
                .= x.v + y.(L.w) by D1,FUNCT_1:13
                .= x.v + x.w  by D1,FUNCT_1:13;
   end;
   for v being VECTOR of X, r being Real holds x.(r*v) = r*x.v
   proof
    let v be VECTOR of X, r be Real;
    thus x.(r*v) = y.(L.(r*v)) by D1,FUNCT_1:13
                .= y.(r*L.v) by LOPBAN_1:def 5
                .= r*y.(L.v) by HAHNBAN:def 3
                .= r*x.v by D1,FUNCT_1:13;
   end; then
   reconsider x as linear-Functional of X by P6,HAHNBAN:def 2,def 3;
   consider N be Real such that
P7:  0 <= N &
     for v be VECTOR of Y holds |. y.v .| <= N * ||. v .||
        by DUALSP01:def 9;
   for v be VECTOR of X holds |. x.v .| <= (M*N) * ||. v .||
   proof
    let v be VECTOR of X;
P8: |.x.v.| = |. y.(L.v) .| by D1,FUNCT_1:13;
P9: |. y.(L.v) .| <= N * ||. (L.v) .|| by P7;
    N*||. (L.v) .|| <= N*(M* ||. v .|| ) by AS1,P7,XREAL_1:64;
    hence thesis by P8,P9,XXREAL_0:2;
   end;
   hence thesis by DUALSP01:def 9,AS1,P7;
end;
