
theorem Th71:
for V be RealNormSpace, x be Point of V st V is non trivial holds
  ( ex X be non empty Subset of REAL st
    X = {|.(Bound2Lipschitz(F,V)).x.|
          where F is Point of DualSp V :||.F.|| = 1 }
    & ||.x.|| = upper_bound X ) &
  ( ex Y be non empty Subset of REAL st
    Y = {|.(Bound2Lipschitz(F,V)).x.|
          where F is Point of DualSp V :||.F.|| <= 1 }
    & ||.x.|| = upper_bound Y )
proof
   let V be RealNormSpace, x be Point of V;
   assume AS: V is non trivial;
   set X = {|.(Bound2Lipschitz(F,V)).x.|
             where F is Point of DualSp V :||.F.|| = 1 };
   set Y = {|.(Bound2Lipschitz(F,V)).x.|
             where F is Point of DualSp V :||.F.|| <= 1 };
   consider F0 be Point of DualSp V such that
P1: ||.F0.|| = 1 by AS,Lm65A;
P2: |.(Bound2Lipschitz(F0,V)).x.| in X
  & |.(Bound2Lipschitz(F0,V)).x.| in Y by P1;
P3: X c= Y
   proof
    let z be object;
    assume z in X; then
    ex F be Point of DualSp V st
     z = |.(Bound2Lipschitz(F,V)).x.| & ||.F.|| = 1;
    hence z in Y;
   end;
P4:Y c= REAL
   proof
    let z be object;
    assume z in Y; then
    ex F be Point of DualSp V st
      z = |.(Bound2Lipschitz(F,V)).x.| & ||.F.|| <= 1;
    hence z in REAL by XREAL_0:def 1;
   end; then
   reconsider Y as non empty Subset of REAL by P2;
   X c= REAL by P3,P4; then
   reconsider X as non empty Subset of REAL by P2;
   per cases;
   suppose X1: x = 0.V;
    for t be object st t in Y holds t in {0 qua Real}
    proof
     let t be object;
     assume t in Y; then
     ex F be Point of DualSp V st
      t = |.(Bound2Lipschitz(F,V)).x.| & ||.F.|| <= 1; then
     t = 0 by ABSVALUE:2,X1,HAHNBAN:20;
     hence t in {0 qua Real} by TARSKI:def 1;
    end; then
P6: Y c= {0 qua Real} & X c= {0 qua Real} by P3;
    ex z be object st z in X by XBOOLE_0:def 1; then
    0 in X by P6,TARSKI:def 1; then
    X = {0 qua Real} & Y = {0 qua Real} by P6,P3,ZFMISC_1:31; then
    upper_bound X = 0 & upper_bound Y = 0 by SEQ_4:9; then
    ||.x.|| = upper_bound X & ||.x.|| = upper_bound Y by X1;
    hence thesis;
   end;
   suppose Z1:x <> 0.V;
X6: for r be ExtReal st r in Y holds r<=||. x .||
    proof
     let r be ExtReal;
     assume r in Y; then
     consider F be Point of DualSp V such that
X4:   r = |.(Bound2Lipschitz(F,V)).x.| & ||.F.|| <= 1;
X5:  |.(Bound2Lipschitz(F,V)).x.|
          <= ||.F.|| * ||.x.|| by DUALSP01:26,SUBSET_1:def 8;
     ||.F.|| * ||.x.|| <= 1 * ||.x.|| by  X4,XREAL_1:64;
     hence r <= ||.x.|| by X4,X5,XXREAL_0:2;
    end; then
    ||.x.|| is UpperBound of Y by XXREAL_2:def 1; then
X7: Y is bounded_above; then
X8: upper_bound X <= upper_bound Y by P3,SEQ_4:48;
    for r be Real st r in Y holds r <= ||. x .|| by X6; then
X9: upper_bound Y <= ||.x.|| by SEQ_4:45; then
X10:upper_bound X <= ||.x.|| by X8,XXREAL_0:2;
    consider F0 be Point of DualSp V such that
Y1:   ||.F0.|| = 1 & (Bound2Lipschitz(F0,V)).x = ||.x.|| by Lm65,Z1;
    |.(Bound2Lipschitz(F0,V)).x.| = ||.x.|| by Y1,ABSVALUE:def 1; then
Y3: ||.x.|| in X by Y1;
    X is bounded_above by P3,X7,XXREAL_2:43; then
    ||.x.|| <= upper_bound X by Y3,SEQ_4:def 1; then
    ||.x.|| = upper_bound X by X10,XXREAL_0:1;
    hence thesis by X9,X8,XXREAL_0:1;
   end;
end;
