reserve s1,s2,q1 for Real_Sequence;
reserve n for Element of NAT;
reserve a,b for Real;

theorem Th9:
for Y be RealNormSpace,
    xseq be FinSequence of Y, yseq be FinSequence of REAL st
    len xseq = len yseq &
    ( for i be Element of NAT st i in dom xseq holds
    ex v be Point of Y st v = xseq/.i & yseq.i = ||.v.|| )
   holds ||.Sum xseq.|| <= Sum yseq
proof
  let Y be RealNormSpace;
  defpred P[Nat] means
    for xseq be FinSequence of Y, yseq be FinSequence of REAL st
     $1=len xseq & len xseq = len yseq &
     ( for i be Element of NAT st i in dom xseq holds
         ex v be Point of Y st v=xseq/.i & yseq.i=||.v.|| )
    holds ||.Sum xseq.|| <= Sum yseq;
A1:P[0]
   proof
    let xseq be FinSequence of Y, yseq be FinSequence of REAL;
    assume A2: 0=len xseq & len xseq = len yseq &
     ( for i be Element of NAT st i in dom xseq holds
        ex v be Point of Y st v = xseq/.i & yseq.i = ||.v.|| ); then
    <*>(the carrier of Y) = xseq;
    then Sum xseq = 0.Y & <*> REAL = yseq by A2,RLVECT_1:43;
    hence thesis by RVSUM_1:72;
   end;

A3:now let i be Nat;
    assume A4: P[i];
    now let xseq be FinSequence of Y, yseq be FinSequence of REAL;
     set xseq0=xseq|i, yseq0=yseq|i;
     assume
A5:   i+1=len xseq & len xseq = len yseq &
      ( for i be Element of NAT st i in dom xseq holds
          ex v be Point of Y st v=xseq/.i & yseq.i=||.v.||);
A6:  for k be Element of NAT st k in dom xseq0 holds
      ex v be Point of Y st v=xseq0/.k & yseq0.k=||.v.||
     proof
      let k be Element of NAT;
      assume A7: k in dom xseq0; then
A8:   k in Seg i & k in dom xseq by RELAT_1:57; then
      consider v be Point of Y such that
A9:    v=xseq/.k & yseq.k=||.v.|| by A5;
      take v;
      xseq/.k = xseq.k by A8,PARTFUN1:def 6; then
      xseq/.k = xseq0.k by A8,FUNCT_1:49;
      hence thesis by A9,A8,A7,PARTFUN1:def 6,FUNCT_1:49;
     end;
     dom xseq = Seg(i+1) by A5,FINSEQ_1:def 3; then
     consider w be Point of Y such that
A10:   w=xseq/.(i+1) & yseq.(i+1)=||.w.|| by A5,FINSEQ_1:4;
A11:  1 <= i + 1 & i + 1 <= len yseq by A5,NAT_1:11;
     yseq = (yseq|i)^<*yseq/.(i+1) *> by A5,FINSEQ_5:21; then
     yseq = yseq0 ^<*(yseq.(i+1))*> by A11,FINSEQ_4:15; then
A12: Sum yseq = Sum yseq0 + yseq.(i+1) by RVSUM_1:74;
A13: i=len xseq0 by A5,FINSEQ_1:59,NAT_1:11; then
A14: ex v be Point of Y st v=xseq/.(len xseq)
        & Sum xseq = Sum xseq0 + v by A5,Lm2;
A15: ||. Sum xseq0 + w.||<= ||.Sum xseq0 .||  + ||. w .|| by NORMSP_1:def 1;
     len xseq0 = len yseq0 by A5,A13,FINSEQ_1:59,NAT_1:11; then
     ||. Sum xseq0 .|| + ||. w  .|| <= Sum yseq0 + yseq.(i+1)
                               by A10,XREAL_1:6,A4,A6,A13;
     hence ||. Sum xseq .|| <= Sum yseq by A5,A10,A12,A14,A15,XXREAL_0:2;
    end;
    hence P[i+1];
   end;
   for i be Nat holds P[i] from NAT_1:sch 2(A1,A3);
   hence thesis;
end;
