 reserve X for RealUnitarySpace;
 reserve x, y, y1, y2 for Point of X;

theorem Th21:
for X being RealUnitarySpace,
    M be non empty Subset of X,
    seq be sequence of X
st rng seq c= the carrier of Ort_Comp(M)
   &
  seq is convergent
holds lim seq in the carrier of Ort_Comp(M)
proof
let X be RealUnitarySpace,
    M be non empty Subset of X,
    seq be sequence of X;
  assume that
A1: rng seq c= the carrier of Ort_Comp(M)
   and
A2: seq is convergent;
now let x be Point of X;
  assume A3:x in M;
defpred P1[Element of NAT,object] means
   $2 =x.|. (seq.$1);
A4: for n being Element of NAT ex y being Element of REAL
    st P1[n,y];
consider xseq be Function of NAT,REAL such that
 A5:for n being Element of NAT holds P1[n,xseq . n]
    from FUNCT_2:sch 3(A4);
reconsider xseq as Real_Sequence;
A6:  for i be Nat holds xseq.i = 0
 proof
  let i be Nat;
   seq.i in rng seq by FUNCT_2:4,ORDINAL1:def 12;
   then
   seq.i in the carrier of Ort_Comp(M) by A1; then
   seq.i in Ort_CompSet(M) by Lm5; then
A7:x .|. (seq.i) = 0 by A3,Def1;
   i in NAT by ORDINAL1:def 12;
   hence thesis by A5,A7;
 end;
for x, y being object st x in dom xseq & y in dom xseq holds
xseq . x = xseq . y
proof
  let x, y be object;
  assume x in dom xseq & y in dom xseq;
  then reconsider i=x,j=y as Nat;
  thus xseq . x = xseq . i
           .= 0 by A6
           .= xseq.j by A6
           .= xseq . y;
end; then
A8:xseq is constant; then
A10: lim xseq = xseq.1 by SEQ_4:26
             .=0 by A6;
for p be Real st 0 < p
ex n be Nat st
      for m be Nat st n<=m holds |. xseq.m - x.|. (lim seq) .| < p
proof
let p be Real;
  assume A11: 0 < p;
   0 + 0 < ||.x.|| + 1 by XREAL_1:8, BHSP_1:28; then
  consider n be Nat such that
     A12: for m be Nat st n<=m holds ||. (seq.m) - lim seq .||
     < p / (||.x.|| + 1) by A2, BHSP_2:19,A11;
  take n;
  let i be Nat;
  assume n<=i; then
   A13: ||. seq.i - lim seq .|| < p / (||.x.|| + 1) by A12;
  |.x .|. (seq.i) - x .|. (lim seq) .|
  = |. x.|. ((seq.i) - lim seq) .| by BHSP_1:12; then
 A14: |.x .|. (seq.i) - x .|. (lim seq) .|
  <= ||.x.|| * ||.(seq.i) - lim seq .|| by BHSP_1:29;
 A15: ||.x.|| + 0 < ||.x.|| + 1 by XREAL_1:8;
 0 <= ||.(seq.i) - lim seq .|| by BHSP_1:28; then
   ||.x.|| * ||.(seq.i) - lim seq .||
    <= ( ||.x.|| + 1) * ||.(seq.i) - lim seq .|| by XREAL_1:64,A15;
 then
 A16: |.x .|. (seq.i) - x .|. (lim seq) .|
     <= ( ||.x.|| + 1) * ||.(seq.i) - lim seq .|| by A14,XXREAL_0:2;
A17: 0 + 0 < ||.x.|| + 1 by XREAL_1:8,BHSP_1:28; then
 ||. seq.i - lim seq .||* ( ||.x.|| + 1)
     < ( p/ (||.x.|| + 1) ) * ( ||.x.|| + 1) by A13,XREAL_1:68;
then
 ||. seq.i - lim seq .||* ( ||.x.|| + 1) < p by A17,XCMPLX_1:87;
then
A18: |.x .|. (seq.i) - x .|. (lim seq) .| < p by A16,XXREAL_0:2;
 i in NAT by ORDINAL1:def 12;
hence thesis by A18,A5;
end;
hence x.|. (lim seq) = 0 by A10,A8,SEQ_2:def 7;
end; then
(lim seq) in Ort_CompSet(M) by Def1;
hence lim seq in the carrier of Ort_Comp(M) by Lm5;
end;
