
theorem Th63:
for V be RealNormSpace, X be SubRealNormSpace of V,
    x0 be Point of V, d be Real st
    ex Z be non empty Subset of REAL
      st Z = {||.x-x0.|| where x is Point of V : x in X} &
         d = lower_bound Z > 0 holds
   not x0 in X &
   ex G be Point of DualSp V st
   ( for x be Point of V st x in X
       holds (Bound2Lipschitz(G,V)).x = 0 )
  & (Bound2Lipschitz(G,V)).x0 = 1 & ||.G.|| = 1/d
proof
   let V be RealNormSpace, X be SubRealNormSpace of V,
       x0 be Point of V, d be Real;
   assume ex Z be non empty Subset of REAL
      st Z = {||.x-x0.|| where x is Point of V : x in X}
       & d = lower_bound Z > 0; then
   consider Z be non empty Subset of REAL such that
AS2:Z = {||.x-x0.|| where x is Point of V : x in X}
  & d = lower_bound Z > 0;
   set M0 = {z+a*x0 where z is Point of V, a is Real : z in X};
A1:0.V = 0.V + 0 * x0 by RLVECT_1:10;
   0.V = 0.X by DUALSP01:def 16; then
D1:0.V in X; then
   0.V in M0 by A1; then
   reconsider M0 as non empty set;
   now let x be object;
    assume x in M0; then
    ex z be Point of V, a be Real st x = z+a*x0 & z in X;
    hence x in the carrier of V;
   end; then
   M0 c= the carrier of V; then
   reconsider M0 as non empty Subset of V;
B0:X is Subspace of V by NORMSP_3:27;
   set M = NLin(M0);
AD1: M0 is linearly-closed
   proof
A0: for v,u be VECTOR of V st v in M0 & u in M0 holds v+u in M0
    proof
     let v,u be VECTOR of V;
     assume A1: v in M0 & u in M0; then
     consider z1 be Point of V, a be Real such that
A3:   v = z1+a*x0 & z1 in X;
     consider z2 be Point of V, b be Real such that
A5:   u = z2+b*x0 & z2 in X by A1;
A7:  v+u = z1 + (a*x0 + (z2 + b*x0)) by A3,A5,RLVECT_1:def 3
        .= z1 + (z2 + (a*x0 + b*x0)) by RLVECT_1:def 3
        .= (z1+z2) + (a*x0+b*x0) by RLVECT_1:def 3
        .= (z1+z2) + (a+b)*x0 by RLVECT_1:def 6;
     z1+z2 in X by B0,A3,A5,RLSUB_1:20;
     hence thesis by A7;
    end;
    for r be Real, v be VECTOR of V st v in M0 holds r*v in M0
    proof
     let r be Real;
     let v be VECTOR of V;
     assume v in M0; then
     consider z be Point of V, a be Real such that
A9:   v = z+a*x0 & z in X;
A11: r*v = r*z + r*(a*x0) by A9,RLVECT_1:def 5
        .= r*z + (r*a)*x0 by RLVECT_1:def 7;
     r*z in X by B0,A9,RLSUB_1:21;
     hence thesis by A11;
    end;
    hence thesis by A0;
   end; then
X01:the carrier of M = M0 by NORMSP_3:31;
V2:x0 = 0.V + 1*x0 by RLVECT_1:def 8; then
V21: x0 in M by D1,X01;
AD2: for v be Point of M
     ex x be Point of V, a be Real st v = x+a*x0 & x in X
   proof
    let v be Point of M;
    v in the carrier of Lin(M0); then
    v in M0 by AD1,NORMSP_3:31;
    hence thesis;
   end;
   reconsider r0=0 as Real;
   for r be ExtReal st r in Z holds r0 <= r
   proof
    let r be ExtReal;
    assume r in Z; then
    ex x be Point of V st r = ||.x-x0.|| & x in X by AS2;
    hence r0 <= r;
   end; then
   r0 is LowerBound of Z by XXREAL_2:def 2; then
U2:Z is bounded_below;
P4:now assume Q1: x0 in X;
    reconsider x0 as Point of V;
    ||.x0 - x0.|| = ||. 0.V .|| by RLVECT_1:15; then
    r0 in Z by Q1,AS2;
    hence contradiction by AS2,U2,SEQ_4:def 2;
   end;
   hence not x0 in X;
AD3: for x1,x2 be Point of V, a1,a2 be Real
    st x1 in X & x2 in X & x1+a1*x0 = x2+a2*x0
     holds x1=x2 & a1=a2
   proof
    let x1,x2 be Point of V, a1,a2 be Real;
    assume P1: x1 in X & x2 in X & x1+a1*x0 = x2+a2*x0; then
    x1 + a1*x0 - x2
               = (x2 + -x2) + a2*x0 by RLVECT_1:def 3
              .= 0.V + a2*x0 by RLVECT_1:5; then
P5: x1 + a1*x0 - x2 - a1*x0 = (a2 - a1)*x0 by RLVECT_1:35;
P6: x1 + a1*x0 - x2 - a1*x0
              = (x1 + a1*x0) + (-x2 - a1*x0) by RLVECT_1:def 3
              .= x1 + (a1*x0 + (-x2 - a1*x0)) by RLVECT_1:def 3
              .= x1 + ((a1*x0 + -x2) - a1*x0) by RLVECT_1:def 3
              .= x1 + (-x2 + (a1*x0 - a1*x0)) by RLVECT_1:def 3
              .= x1 + (-x2 + 0.V) by RLVECT_1:15;
P7: a2 = a1
    proof
     assume a2 <> a1; then
Q0:  a2 - a1 <> 0; then
Q1:  (a2-a1)*(1/(a2-a1)) = 1 by XCMPLX_1:106;
     1*(x1 - x2) = (a2 - a1)*x0 by P5,P6,RLVECT_1:def 8; then
     (a2-a1)*((1/(a2-a1))*(x1-x2)) = (a2-a1)*x0 by Q1,RLVECT_1:def 7; then
Q2:  (1/(a2-a1))*(x1-x2) = x0 by Q0,RLVECT_1:36;
     reconsider r=1/(a2-a1) as Real;
Q4:  r*x1 in X & r*x2 in X by B0,P1,RLSUB_1:21;
     r*(x1-x2) = r*x1-r*x2 by RLVECT_1:34;
     hence contradiction by P4,Q2,B0,Q4,RLSUB_1:23;
    end; then
    x1 - x2 = 0.V by P5,P6,RLVECT_1:10;
    hence thesis by P7,RLVECT_1:21;
   end;
   defpred P[object,object] means
    ex z be Point of V, a be Real st z in X & $1 = z+a*x0 & $2 = a;
F1:for v being Element of M
     ex a being Element of REAL st P[v,a]
   proof
    let v be Element of M;
    consider z be Point of V, a be Real such that
B0:  v = z+a*x0 & z in X by AD2;
    reconsider aa=a as Element of REAL by XREAL_0:def 1;
    take aa;
    thus thesis by B0;
   end;
   consider f being Function of M,REAL such that
A1F:for x being Element of M holds P[x,f.x] from FUNCT_2:sch 3(F1);
A1:for v be Point of M, z be Point of V, a be Real st
     z in X & v= z+a*x0 holds f.v = a
   proof
    let v be Point of M, z be Point of V, a be Real;
    assume AS0: z in X & v = z+a*x0;
    ex z1 be Point of V, a1 be Real st
     z1 in X & v = z1+a1*x0 & f.v = a1 by A1F;
    hence f.v = a by AS0,AD3;
   end;
   f is linear-Functional of M
   proof
B1: f is additive
    proof
     let v,w be VECTOR of M;
     consider x1 be Point of V, a1 be Real such that
B11:  v = x1+a1*x0 & x1 in X by AD2;
     consider x2 be Point of V, a2 be Real such that
B13:  w = x2+a2*x0 & x2 in X by AD2;
B14: f.v = a1 & f.w = a2 by A1,B11,B13;
     v+w = (x1 + a1*x0) + (x2 + a2*x0) by B11,B13,NORMSP_3:28
        .= x1 + (a1*x0 + (x2 + a2*x0)) by RLVECT_1:def 3
        .= x1 + (x2 + (a1*x0 + a2*x0)) by RLVECT_1:def 3
        .= (x1+x2) + (a1*x0+a2*x0) by RLVECT_1:def 3
        .= (x1+x2) + (a1+a2)*x0 by RLVECT_1:def 6;
     hence f.(v+w) = f.v + f.w by B14,A1,B0,B11,B13,RLSUB_1:20;
    end;
    f is homogeneous
    proof
     let v be VECTOR of M, r be Real;
     consider x be Point of V, a be Real such that
B11:  v = x+a*x0 & x in X by AD2;
     r*v = r*(x+a*x0) by B11,NORMSP_3:28
        .= r*x + r*(a*x0) by RLVECT_1:def 5
        .= r*x + (r*a)*x0 by RLVECT_1:def 7;
     hence f.(r*v) = r*a by A1,B0,B11,RLSUB_1:21
                  .= r*f.v by A1,B11;
    end;
    hence thesis by B1;
   end; then
   reconsider f as linear-Functional of M;
A5:for v be Point of M holds |.f.v.| <= (1/d)*||.v.||
   proof
    let v be Point of M;
    consider x be Point of V, a be Real such that
B5:   v = x+a*x0 & x in X by AD2;
    per cases;
    suppose a = 0; then
     |.f.(x+a*x0).| = 0 by A1,B5,ABSVALUE:2;
     hence |.f.v.| <= (1/d)*||.v.|| by AS2,B5;
    end;
    suppose B6: a <> 0;
C3:  ||.x+a*x0.|| = ||.1*x + a*x0.|| by RLVECT_1:def 8
                 .= ||.(a*(1/a))*x + a*x0.|| by B6,XCMPLX_1:106
                 .= ||.a*((1/a)*x) + a*x0.|| by RLVECT_1:def 7
                 .= ||.a*((1/a)*x + x0).|| by RLVECT_1:def 5
                 .= |.a.|*||.(1/a)*x + x0.|| by NORMSP_1:def 1;
C4:  ||.(1/a)*x + x0.|| = ||.-((1/a)*x + x0).|| by NORMSP_1:2
                       .= ||.-(1/a)*x -x0.|| by RLVECT_1:30;
     set s = ||.-(1/a)*x-x0.||;
C52: -(1/a)*x = (1/a)*(-x) by RLVECT_1:25;
     -x in X by B0,B5,RLSUB_1:22; then
     -(1/a)*x in X by B0,C52,RLSUB_1:21; then
     s in Z by AS2; then
C5:  ||.-(1/a)*x-x0.|| >= d by AS2,U2,SEQ_4:def 2;
     |.a.| >= 0 by COMPLEX1:46; then
     |.a.|*||.-(1/a)*x -x0.|| >= |.a.|*d by C5,XREAL_1:64; then
     ||.x+a*x0.||/d >= |.a.| by AS2,C3,C4,XREAL_1:77; then
     (1/d)*||.x+a*x0.|| >= |.a.| by XCMPLX_1:99; then
     |.f.(x+a*x0).| <= (1/d)* ||.x+a*x0.|| by A1,B5;
     hence |.f.v.| <= (1/d)*||.v.|| by B5,NORMSP_3:28;
    end;
   end; then
   f is Lipschitzian by AS2; then
   reconsider f as Lipschitzian linear-Functional of M;
   reconsider F=f as Point of DualSp M by DUALSP01:def 10;
   consider g be Lipschitzian linear-Functional of V, G be Point of DualSp V
    such that
C1: g = G & g| (the carrier of M) = f & ||.G.||=||.F.|| by DUALSP01:36;
A31:for x be Point of V st x in X holds (Bound2Lipschitz(G,V)).x = 0
   proof
    let x be Point of V;
    assume A32: x in X;
    x = x + 0.V; then
A33:x = x + 0 * x0 by RLVECT_1:10; then
A34:x in M by X01,A32;
    thus (Bound2Lipschitz(G,V)).x = G.x by SUBSET_1:def 8
                            .= f.x by A34,C1,FUNCT_1:49
                            .= 0 by A33,A32,A34,A1;
   end;
A12:
   (Bound2Lipschitz(G,V)).x0 = G.x0 by SUBSET_1:def 8
                            .= f.x0 by C1,V21,FUNCT_1:49
                            .= 1 by A1,V2,D1,V21;
   take G;
   now let r be Real;
    assume r in PreNorms f; then
    consider t be VECTOR of M such that
C1:  r = |.f.t.| & ||.t.|| <= 1;
C3: |.f.t.| <= (1/d)*||.t.|| by A5;
    (1/d) * ||.t.|| <= (1/d) * 1 by AS2,C1,XREAL_1:64;
    hence r <= (1/d) by C1,C3,XXREAL_0:2;
   end; then
   upper_bound PreNorms f <= (1/d) by SEQ_4:45; then
B3: ||.F.|| <= 1/d by DUALSP01:24;
   now let s be Real;
    assume 0 < s; then
    consider r be Real such that
B32: r in Z & r < (lower_bound Z) + s by U2,SEQ_4:def 2;
    consider x be Point of V such that
B34: r = ||.x-x0.|| & x in X by B32,AS2;
B35:x-x0 = x+(-1)*x0 by RLVECT_1:16; then
    x-x0 in M0 by B34; then
    reconsider xx0=x-x0 as Point of M by AD1,NORMSP_3:31;
    |. f.xx0 .| = |. -1 .| by B35,A1,B34
               .= |. 1 .| by COMPLEX1:52
               .= 1 by COMPLEX1:43; then
B38:1 <= ||.F.|| * ||.xx0.|| by DUALSP01:26;
    ||.xx0.|| = r by NORMSP_3:28,B34; then
    ||.F.|| * ||.xx0.|| <= ||.F.|| * (d+s) by AS2,B32,XREAL_1:64;
    hence 1 <= ||.F.|| * d + ||.F.|| * s by B38,XXREAL_0:2;
   end; then
   1 <= ||.F.|| * d by NORMSP_3:22; then
   1/d <= (||.F.|| *d) /d by XREAL_1:72,AS2; then
   1/d <= ||.F.|| by XCMPLX_1:89,AS2;
   hence thesis by A31,A12,C1,B3,XXREAL_0:1;
end;
