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 Th14:
  rseq is convergent & seq is convergent implies lim (rseq (#) seq
  ) = (lim rseq) * (lim seq)
proof
  assume that
A1: rseq is convergent and
A2: seq is convergent;
  set g2= lim seq;
  reconsider g1 = lim rseq as Real;
  set g=g1*g2;
  rseq is bounded by A1,SEQ_2:13;
  then consider r be Real such that
A3: 0<r and
A4: for n be Nat holds |.rseq.n.|<r by SEQ_2:3;
  reconsider r as Real;
A5: 0+0<||.g2.||+r by A3,NORMSP_1:4,XREAL_1:8;
A6: 0<=||.g2.|| by NORMSP_1:4;
A7: for p be Real st 0<p
  ex n be Nat st for m be Nat
  st n <=m holds ||.((rseq(#)seq).m)-g.||<p
  proof
    let p be Real;
    assume 0<p;
    then
A8: 0<p/(||.g2.||+r) by A5,XREAL_1:139;
    then consider n1 be Nat such that
A9: for m be Nat st n1<=m holds |.rseq.m-g1.|<p/(||.g2.||
    +r) by A1,SEQ_2:def 7;
    consider n2 be Nat such that
A10: for m be Nat  st n2<=m holds ||.seq.m-g2.||<p/(||.g2.||
    +r) by A2,A8,NORMSP_1:def 7;
    take n=n1+n2;
    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;
    ||.(rseq.m-g1)*g2.||=||.g2.||*|.rseq.m-g1.| by NORMSP_1:def 1;
    then
A13: ||.(rseq.m-g1)*g2.||<=||.g2.||*(p/(||.g2.||+r)) by A6,A12,XREAL_1:64;
A14: 0<=|.rseq.m.| & 0<=||.seq.m-g2.|| by COMPLEX1:46,NORMSP_1:4;
    n2<=n by NAT_1:12;
    then n2<=m by A11,XXREAL_0:2;
    then
A15: ||.seq.m-g2.||<p/(||.g2.||+r) by A10;
    ||.((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
A16: ||.((rseq(#)seq).m)-g.||<= ||.rseq.m*(seq.m-g2).||+||.(rseq.m-g1)*g2
    .|| by NORMSP_1:def 1;
    |.rseq.m.|<r by A4;
    then |.rseq.m.|*||.seq.m-g2.||<r*(p/(||.g2.||+r)) by A14,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 A5,XCMPLX_1:87;
    then ||.rseq.m*(seq.m-g2).||+||.(rseq.m-g1)*g2.||<p by A17,A13,XREAL_1:8;
    hence thesis by A16,XXREAL_0:2;
  end;
  (rseq (#) seq) is convergent by A1,A2,Th13;
  hence thesis by A7,NORMSP_1:def 7;
end;
