reserve n,k for Element of NAT;
reserve x,y,X for set;
reserve g,r,p for Real;
reserve S for RealNormSpace;
reserve rseq for Real_Sequence;
reserve seq,seq1 for sequence of S;
reserve x0 for Point of S;
reserve Y for Subset of S;

theorem Th13:
  rseq is convergent & seq is convergent implies rseq (#) seq is convergent
proof
  assume that
A1: rseq is convergent and
A2: seq is convergent;
  consider g1 be Real such that
A3: for p be Real st 0<p ex n be Nat st for m be
  Nat st n<=m holds |.rseq.m-g1.|<p by A1,SEQ_2:def 6;
  consider g2 be Point of S such that
A4: for p be Real st 0<p
   ex n be Nat st for m be Nat st n<=m holds ||.seq.m-g2.||<p
    by A2;
  reconsider g1 as Real;
  take g=g1*g2;
  let p;
  rseq is bounded by A1,SEQ_2:13;
  then consider r be Real such that
A5: 0<r and
A6: for n be Nat holds |.rseq.n.|<r by SEQ_2:3;
  reconsider r as Real;
A7: 0+0<||.g2.||+r by A5,NORMSP_1:4,XREAL_1:8;
  assume
A8: 0<p;
  then consider n1 be Nat such that
A9: for m be Nat st n1<=m holds |.rseq.m-g1.|<p/(||.g2.||+r
  ) by A3,A7,XREAL_1:139;
  consider n2 be Nat such that
A10: for m be Nat st n2<=m holds ||.seq.m-g2.||<p/(||.g2.||+r
  ) by A4,A7,A8,XREAL_1:139;
   reconsider n=n1+n2 as Nat;
  take n;
  let m be Nat such that
A11: n<=m;
  n1<=n1+n2 by NAT_1:12;
  then n1<=m by A11,XXREAL_0:2;
  then
A12: |.rseq.m-g1.|<=p/(||.g2.||+r) by A9;
  0<=||.g2.|| & ||.(rseq.m-g1)*g2.||=||.g2.||*|.rseq.m-g1.| by NORMSP_1:4
,def 1;
  then
A13: ||.(rseq.m-g1)*g2.||<=||.g2.||*(p/(||.g2.||+r)) by A12,XREAL_1:64;
  ||.((rseq(#)seq).m)-g.|| =||.rseq.m*seq.m-g1*g2.|| by Def2
    .=||.rseq.m*seq.m-0.S-g1*g2.|| by RLVECT_1:13
    .=||.rseq.m*seq.m-(rseq.m*g2-rseq.m*g2)-g1*g2.|| by RLVECT_1:15
    .=||.(rseq.m*seq.m-rseq.m*g2+rseq.m*g2)-g1*g2.|| by RLVECT_1:29
    .=||.rseq.m*(seq.m-g2)+rseq.m*g2-g1*g2.|| by RLVECT_1:34
    .=||.rseq.m*(seq.m-g2)+(rseq.m*g2-g1*g2).|| by RLVECT_1:def 3
    .=||.rseq.m*(seq.m-g2)+(rseq.m-g1)*g2.|| by RLVECT_1:35;
  then
A14: ||.((rseq(#)seq).m)-g.||<= ||.rseq.m*(seq.m-g2).||+||.(rseq.m-g1)*g2.||
  by NORMSP_1:def 1;
  n2<=n by NAT_1:12;
  then n2<=m by A11,XXREAL_0:2;
  then
A15: ||.seq.m-g2.||<p/(||.g2.||+r) by A10;
A16: 0<=|.rseq.m.| & 0<=||.seq.m-g2.|| by COMPLEX1:46,NORMSP_1:4;
  |.rseq.m.|<r by A6;
  then |.rseq.m.|*||.seq.m-g2.||<r*(p/(||.g2.||+r)) by A16,A15,XREAL_1:96;
  then
A17: ||.rseq.m*(seq.m-g2).||<r*(p/(||.g2.||+r)) by NORMSP_1:def 1;
  r*(p/(||.g2.||+r))+||.g2.||*(p/(||.g2.||+r)) =(p/(||.g2.||+r))*(||.g2 .||+r)
    .=p by A7,XCMPLX_1:87;
  then ||.rseq.m*(seq.m-g2).||+||.(rseq.m-g1)*g2.||<p by A17,A13,XREAL_1:8;
  hence thesis by A14,XXREAL_0:2;
end;
