 reserve S, T for RealNormSpace;
 reserve F for Subset of Funcs(the carrier of S,the carrier of T);
 reserve S,Z for RealNormSpace;
 reserve T for RealBanachSpace;
 reserve F for Subset of Funcs(the carrier of S,the carrier of T);

theorem Th13:
  for M be non empty MetrSpace,S be non empty compact TopSpace,
      T be NormedLinearTopSpace
   st S = TopSpaceMetr(M) & T is complete holds
  for G be Subset of Funcs(the carrier of M, the carrier of T),
      H be non empty Subset of
    MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T) st
     G = H & (MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T))
       | H is totally_bounded holds
    G is equibounded & G is equicontinuous
proof
  let M be non empty MetrSpace,S be non empty compact TopSpace,
      T be NormedLinearTopSpace;
  assume A1:  S = TopSpaceMetr(M) & T is complete;
  let G be Subset of Funcs(the carrier of M, the carrier of T),
      H be non empty Subset of
    (MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T));
  assume A2: G = H;
  set Z = R_NormSpace_of_ContinuousFunctions(S,T);
  set MZH = (MetricSpaceNorm Z) | H;
A3:the carrier of MZH = H by TOPMETR:def 2;
  assume
A4:  (MetricSpaceNorm Z) | H is totally_bounded;
  consider L being Subset-Family of MZH such that
A5: L is finite & the carrier of MZH = union L
 & for C being Subset of MZH st C in L holds
     ex w being Element of MZH st C = Ball (w,1) by A4;
defpred P1[object,object] means
   ex w be Point of MZH st $2 = w & $1 =Ball(w,1);
A6: for D be object st D in L
     ex w be object st w in the carrier of MZH & P1[D,w]
  proof
   let D be object;
   assume A7: D in L; then
   reconsider D0=D as Subset of MZH;
   consider w being Element of MZH such that
A8:D0 = Ball (w,1) by A5,A7;
   take w;
   thus w in the carrier of MZH & P1[D,w] by A8;
end;
  consider U being Function of L,the carrier of MZH such that
  A9: for D being object st D in L
        holds P1[D,U.D] from FUNCT_2:sch 1(A6);
A10:for D being object st D in L holds D = Ball(U/.D,1)
proof
let D be object;
assume A11: D in L; then
A12:  ex x0 be Point of MZH st U.D = x0 & D = Ball(x0,1) by A9;
dom U = L by FUNCT_2:def 1;
hence D = Ball(U/.D,1) by A12,A11,PARTFUN1:def 6;
end;
set NF = the normF of Z;
A13: dom U = L by FUNCT_2:def 1;
reconsider NFU = NF.: (rng U) as finite Subset of REAL by A5;
consider xx be object such that
A14: xx in L by XBOOLE_0:def 1,A5,ZFMISC_1:2;
rng U <> {} by A13,A14,FUNCT_1:3;
then consider xx be object such that
A15: xx in rng U by XBOOLE_0:def 1;
reconsider xx as Point of MZH by A15;
set d0 = upper_bound NFU;
A16: for r be Real st r in NFU holds r <= d0 by SEQ_4:def 1;
  set K = d0 + 1;
 for f be Function of the carrier of M,the carrier of T
             st f in G holds for x be Element of M
               holds ||.f.x.|| <= K
proof
  let f be Function of the carrier of M,the carrier of T;
  assume f in G; then
  consider C be set such that
A17: f in C & C in L by A5,TARSKI:def 4,A3,A2;
  reconsider C as Subset of the carrier of MZH by A17;
  reconsider pf=f as Element of MZH by A17;
A18: pf in H by A3;
  set pg = U/.C;
  reconsider pg as Element of MZH;
  A19:C = Ball (pg,1) by A10,A17;
  A20: pg in H by A3; then
  pg in Z; then
  ex f be Function of the carrier of S, the carrier of T
    st pg=f & f is continuous; then
  reconsider g=pg as Function of S,T;
  pf in { y where y is Point of MZH : dist(pg,y) < 1 }
    by METRIC_1:def 14,A17,A19; then
  A21:ex y be Point of MZH st pf=y & dist(pg,y) < 1;
  reconsider ppf=pf,ppg=pg as Element of MetricSpaceNorm Z by A18,A20;
  A22:  dist(ppg,ppf) < 1 by A21,TOPMETR:def 1;
  reconsider pppf=ppf,pppg=ppg as Point of Z;
  A23: ||.pppg-pppf.|| < 1 by A22,NORMSP_2:def 1;
  pppf = pppf-pppg + pppg by RLVECT_4:1; then
  ||.pppf.|| <= ||.pppf-pppg .|| + ||.pppg.|| by NORMSP_1:def 1; then
A24: ||.pppf.|| <= ||.pppg-pppf.|| + ||.pppg.|| by NORMSP_1:7;
A25:C in dom U by FUNCT_2:def 1,A17;
  U.C in rng U by FUNCT_1:3,A13,A17; then
  pg in rng U by A25,PARTFUN1:def 6; then
  ||.pppg.|| <= d0 by A16, FUNCT_2:35; then
  ||.pppg-pppf.|| + ||.pppg.|| <= 1+d0 by XREAL_1:8,A23; then
A26: ||.pppf.|| <= K by A24,XXREAL_0:2;
  let x be Element of M;
   reconsider x0 = x as Point of S by A1;
   reconsider f0=f
    as Function of the carrier of S,the carrier of T by A1;
   ||.f0.x0.|| <=||.pppf.|| by C0SP3:37;
   hence ||.f.x.|| <= K by A26,XXREAL_0:2;
end;
 hence G is equibounded;
for e be Real st 0 < e
   ex d be Real st 0 < d &
    for f be Function of the carrier of M,the carrier of T
             st f in G
      holds
      for x1,x2 be Point of M
          st dist(x1,x2) < d holds ||.f.x1-f.x2.|| < e
proof
  let e be Real;
  assume A27:0 < e; then
  consider L being Subset-Family of MZH such that
A28: L is finite & the carrier of MZH = union L
 & for C being Subset of MZH st C in L holds
     ex w being Element of MZH st C = Ball (w,e/3) by A4;
  defpred P1[object,object] means
   ex w be Point of MZH st $2 = w & $1 = Ball(w,e/3);
A29: for D be object st D in L
     ex w be object st w in the carrier of MZH & P1[D,w]
  proof
    let D be object;
    assume A30: D in L; then
    reconsider D0=D as Subset of MZH;
    consider w being Element of MZH such that
    A31: D0 = Ball (w,e/3) by A28,A30;
    take w;
    thus w in the carrier of MZH & P1[D,w] by A31;
end;
  consider U being Function of L,the carrier of MZH such that
  A32: for D being object
         st D in L
        holds P1[D,U.D] from FUNCT_2:sch 1(A29);
A33:for D being object st D in L holds D = Ball(U/.D,e/3)
proof
let D be object;
assume A34: D in L; then
A35:   ex x0 be Point of MZH st U.D = x0 & D = Ball(x0,e/3) by A32;
dom U = L by FUNCT_2:def 1;
  hence D = Ball(U/.D,e/3) by A35,A34,PARTFUN1:def 6;
end;
  defpred P1[Element of MZH,Real] means
     ex f be Function of the carrier of S,the carrier of T
       st $1=f & 0 < $2
       & for x1,x2 be Point of M st dist(x1,x2) < $2
     holds ||.f/.x1-f/.x2.|| < e/3;
  A36: for x0 being Element of the carrier of MZH
        ex d being Element of REAL st P1[x0,d]
  proof
   let x0 be Element of the carrier of MZH;
   x0 in H by A3; then
   x0 in Z; then
   A37: ex f be Function of the carrier of S, the carrier of T
  st x0=f & f is continuous; then
   reconsider f=x0 as Function of the carrier of S,
      the carrier of T;
   consider d be Real such that
   A38: 0 < d &
     for x1,x2 be Point of M st dist(x1,x2) < d
     holds ||.f/.x1-f/.x2.|| < e/3 by A27,Th4,A1,A37;
  reconsider d0=d as Element of REAL by XREAL_0:def 1;
   take d0;
   thus thesis by A38;
end;
  consider NF being Function of the carrier of MZH,REAL such that
A39: for f being Element of the carrier of MZH
        holds P1[f,NF.f] from FUNCT_2:sch 3(A36);
A40: dom U = L by FUNCT_2:def 1;
reconsider NFU = NF.: (rng U) as finite Subset of REAL by A28;
consider xx be object such that
A41: xx in L by XBOOLE_0:def 1,A28,ZFMISC_1:2;
rng U <> {} by A40,A41,FUNCT_1:3; then
consider xx be object such that
A42: xx in rng U by XBOOLE_0:def 1;
reconsider xx as Point of MZH by A42;
A43: NFU is bounded_below & lower_bound NFU in NFU by SEQ_4:133,A42;
set d = lower_bound NFU;
consider xx being object such that
 A44: xx in dom NF & xx in rng U & d = NF . xx by FUNCT_1:def 6,A43;
reconsider xx as Point of MZH by A44;
A45:ex f be Function of the carrier of S,the carrier of T
   st xx=f & 0 < NF.xx
           & for x1,x2 be Point of M st dist(x1,x2) < NF.xx
     holds ||.f/.x1-f/.x2.|| < e/3 by A39;
take d;
thus 0 < d by A45,A44;
thus for f be Function of the carrier of M,the carrier of T
             st f in G holds
      for x1,x2 be Point of M
          st dist(x1,x2) < d holds ||.f.x1-f.x2.|| < e
proof
let f0 be Function of the carrier of M,the carrier of T;
  assume f0 in G; then
     consider C be set such that
      A46: f0 in C & C in L by A28,TARSKI:def 4,A3,A2;
  reconsider f=f0 as Function of the carrier of S,the carrier of T by A1;
reconsider C as Subset of the carrier of MZH by A46;
reconsider pf=f0 as Element of MZH by A46;
A47: pf in H by A3;
reconsider pg = U/.C as Element of MZH;
A48:C = Ball (pg,e/3) by A33,A46;
A49: pg in H by A3; then
pg in ContinuousFunctions(S,T); then
ex f be Function of the carrier of S, the carrier of T
  st pg=f & f is continuous; then
reconsider g=pg as Function of S,T;
pf in { y where y is Point of MZH : dist(pg,y) < e/3 }
  by METRIC_1:def 14,A46,A48; then
A50:ex y be Point of MZH st pf=y & dist(pg,y) < e/3;
  reconsider ppf=pf,ppg=pg as Element of MetricSpaceNorm Z by A47,A49;
A51:  dist(ppg,ppf) < e/3 by A50,TOPMETR:def 1;
  reconsider pppf=ppf,pppg=ppg as Point of Z;
A52: ||.pppg-pppf.|| < e/3 by A51,NORMSP_2:def 1;
A53:C in dom U by FUNCT_2:def 1,A46;
  U.C in rng U by FUNCT_1:3,A40,A46; then
  pg in rng U by A53,PARTFUN1:def 6; then
A54: NF.pppg in NF.: (rng U) by FUNCT_2:35;
let x1,x2 be Element of M;
   reconsider x10 = x1,x20 = x2 as Point of S by A1;
   assume A55: dist(x1,x2) < d;
   d <=NF.pg by A54,SEQ_4:def 2; then
A56: dist(x1,x2) < NF.pg by A55,XXREAL_0:2;
ex f be Function of the carrier of S,the carrier of T
   st pg=f & 0 < NF.pg
           & for x1,x2 be Point of M st dist(x1,x2) < NF.pg
     holds ||.f/.x1-f/.x2.|| < e/3 by A39; then
A57: ||.g/.x1-g/.x2.|| < e/3 by A56;
pppg-pppf in ContinuousFunctions(S,T); then
 ex f be Function of the carrier of S, the carrier of T
  st pppg-pppf=f & f is continuous; then
reconsider gf=pppg-pppf as Function of S,T;
pppf-pppg in ContinuousFunctions(S,T); then
 ex f be Function of the carrier of S, the carrier of T
  st pppf-pppg=f & f is continuous; then
reconsider fg=pppf-pppg as Function of S,T;
   ||.gf.x20.|| <= ||.pppg-pppf.|| by C0SP3:37; then
A59: ||.g.x20 - f.x20 .|| <= ||.pppg-pppf.|| by C0SP3:48;
f/.x20=f.x20 & g/.x20 =g.x20; then
A60: ||.g/.x2-f/.x2.|| < e/3 by A59,XXREAL_0:2,A52;
  ||.fg.x10.|| <= ||.pppf-pppg.|| by C0SP3:37; then
  ||.f.x10 - g.x10 .|| <= ||.pppf-pppg.|| by C0SP3:48; then
A62: ||.f.x10 - g.x10 .|| <= ||.pppg-pppf.|| by NORMSP_1:7;
  f/.x10=f.x10 & g/.x10 =g.x10; then
A63: ||.f/.x1-g/.x1.|| < e/3 by A62,XXREAL_0:2,A52;
f/.x1-f/.x2 = f/.x1-g/.x1+g/.x1-f/.x2 by RLVECT_4:1
      .= f/.x1-g/.x1+ (g/.x1-f/.x2) by RLVECT_1:28
      .= f/.x1-g/.x1+ (g/.x1-g/.x2+ g/.x2 -f/.x2) by RLVECT_4:1
      .= f/.x1-g/.x1+ (g/.x1-g/.x2+ (g/.x2 -f/.x2)) by RLVECT_1:28; then
A64:||.f/.x1-f/.x2.|| <= ||.f/.x1-g/.x1.||
+ ||.g/.x1-g/.x2+ (g/.x2 -f/.x2).|| by NORMSP_1:def 1;
A65: ||.g/.x1-g/.x2+ (g/.x2 -f/.x2).||
        <= ||.g/.x1-g/.x2.|| + ||.g/.x2 -f/.x2.|| by NORMSP_1:def 1;
 ||.g/.x1-g/.x2.|| + ||.g/.x2 -f/.x2.||
  < e/3 + e/3 by A57,A60,XREAL_1:8; then
  ||.g/.x1-g/.x2+ (g/.x2 -f/.x2).|| < (e/3+e/3) by XXREAL_0:2,A65; then
A66: ||.f/.x1-g/.x1.|| + ||.g/.x1-g/.x2+ (g/.x2 -f/.x2).||
   < e/3 + (e/3+e/3) by A63,XREAL_1:8;
  f/.x10=f.x10 & f/.x20 = f.x20;
  hence ||.f0.x1-f0.x2.|| < e by A66,A64,XXREAL_0:2;
end;
end;
hence G is equicontinuous;
end;
