reserve i,n,m for Nat;

theorem Th17:
for xseq be FinSequence of REAL m, yseq be FinSequence of REAL st
 len xseq = len yseq &
 ( for i be Nat st i in dom xseq holds
    ex v be Element of REAL m st v = xseq.i & yseq.i = |.v .| )
  holds |.Sum xseq.| <= Sum yseq
proof
   defpred P[Nat] means
    for xseq be FinSequence of REAL m, yseq be FinSequence of REAL st
     $1=len xseq & len xseq = len yseq &
     ( for i be Nat st i in dom xseq holds
         ex v be Element of REAL m st v=xseq.i & yseq.i=|.v .| )
    holds |.Sum xseq.| <= Sum yseq;
A1:P[0]
   proof
    let xseq be FinSequence of REAL m, yseq be FinSequence of REAL;
    assume 0=len xseq & len xseq = len yseq &
     ( for i be Nat st i in dom xseq holds
        ex v be Element of REAL m st v = xseq.i & yseq.i = |.v .| ); then
    Sum xseq = 0*m & <*> REAL = yseq by EUCLID_7:def 11;
    hence thesis by EUCLID:7,RVSUM_1:72;
   end;
A2:now let i be Nat;
    assume A3: P[i];
    now let xseq be FinSequence of REAL m, yseq be  FinSequence of REAL;
     set xseq0=xseq|i, yseq0=yseq|i;
     assume
A4:   i+1=len xseq & len xseq = len yseq &
      ( for i be Nat st i in dom xseq holds
          ex v be Element of REAL m st v=xseq.i & yseq.i=|.v .|);
A5:  for k be Nat st k in dom xseq0 holds
      ex v be Element of REAL m st v=xseq0.k & yseq0.k=|.v .|
     proof
      let k be Nat;
      assume k in dom xseq0; then
A6:   k in Seg i & k in dom xseq by RELAT_1:57; then
      consider v be Element of REAL m such that
A7:    v=xseq.k & yseq.k=|.v .| by A4;
      take v;
      thus thesis by A6,A7,FUNCT_1:49;
     end;
     dom xseq = Seg(i+1) by A4,FINSEQ_1:def 3; then
     consider w be Element of REAL m such that
A8:   w=xseq.(i+1) & yseq.(i+1)=|.w .| by A4,FINSEQ_1:4;
A9: 1 <= i + 1 & i + 1 <= len yseq by A4,NAT_1:11;
     yseq = (yseq|i)^<*yseq/.(i+1) *> by A4,FINSEQ_5:21; then
     yseq = yseq0 ^<*(yseq.(i+1))*> by A9,FINSEQ_4:15; then
A10: Sum yseq = Sum yseq0 + yseq.(i+1) by RVSUM_1:74;
A11: i=len xseq0 by A4,FINSEQ_1:59,NAT_1:11; then
A12: ex v be Element of REAL m st v=xseq.(len xseq)
        & Sum xseq = Sum xseq0 + v by A4,Th15;
A13: |. Sum xseq0 + w.|<= |.Sum xseq0 .|  + |. w .| by EUCLID:12;
     len xseq0 = len yseq0 by A4,A11,FINSEQ_1:59,NAT_1:11; then
     |. Sum xseq0 .| <= Sum yseq0 by A3,A5,A11; then
     |. Sum xseq0 .| + |. w  .| <= Sum yseq0 + yseq.(i+1) by A8,XREAL_1:6;
     hence |. Sum xseq .| <= Sum yseq by A4,A8,A10,A12,A13,XXREAL_0:2;
    end;
    hence P[i+1];
   end;
   for i be Nat holds P[i] from NAT_1:sch 2(A1,A2);
   hence thesis;
end;
