 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem
  for F,G be FinSequence of F_Real
  st len F = len G & (for i being Nat st i in dom F holds F.i <= G.i) holds
  Sum F <= Sum G
   proof
     let F,G be FinSequence of F_Real;
     assume
A1:  len F = len G & for i being Nat st i in dom F holds F.i <= G.i;
     Sum F <= Sum G
     proof
       defpred P[Nat] means
       for F1,G1 be FinSequence of F_Real st
       len F1 = $1 & len F1 = len G1 &
       (for i be Nat st i in dom F1 holds F1.i <= G1.i) holds Sum F1 <= Sum G1;
A2:    P[0]
       proof
         let F1,G1 be FinSequence of F_Real;
         assume
A3:      len F1 = 0 & len F1 = len G1 &
         for i be Nat st i in dom F1 holds F1.i <= G1.i;
         Sum F1 <= Sum G1
         proof
           F1 = <*>the carrier of F_Real & G1 = <*>the carrier of F_Real by A3;
           hence thesis;
         end;
         hence thesis;
       end;
A5:    for n be Nat st P[n] holds P[n+1]
       proof
         let n be Nat;
         assume
A6:      P[n];
         for F1,G1 be FinSequence of F_Real st
         len F1 = n+1 & len F1 = len G1 &
         (for i be Nat st i in dom F1 holds F1.i <= G1.i)
         holds Sum F1 <= Sum G1
         proof
           let F1,G1 be FinSequence of F_Real;
           assume
A7:        len F1 = n+1 & len F1 = len G1 &
           for i be Nat st i in dom F1 holds F1.i <= G1.i;
           Sum F1 <= Sum G1
           proof
             reconsider F0 = F1| n as FinSequence of F_Real;
             reconsider G0 = G1| n as FinSequence of F_Real;
             n+1 in Seg (n+1) by FINSEQ_1:4; then
A8:          n+1 in dom F1 & n+1 in dom G1 by A7,FINSEQ_1:def 3;
A9:          len F0 = n & len G0 = n by FINSEQ_1:59,A7,NAT_1:11; then
A10:         dom F0 = Seg n & dom G0 = Seg n by FINSEQ_1:def 3;
             dom F1 = Seg (n+1) & dom G1 = Seg (n+1) by A7,FINSEQ_1:def 3; then
A11:         dom F0 c= dom F1 & dom G0 c= dom G1 by A10,FINSEQ_1:5,NAT_1:11;
             F1.(n+1) in rng F1 by A8,FUNCT_1:3; then
             reconsider Fn1 = F1.(n+1) as Element of F_Real;
             G1.(n+1) in rng G1 by A8,FUNCT_1:3; then
             reconsider Gn1 = G1.(n+1) as Element of F_Real;
             F1 = F0^<* F1/.(len F1) *> by FINSEQ_5:21,A7
             .= F0^<* Fn1 *> by A7,A8,PARTFUN1:def 6; then
A14:         Sum F1 = Sum F0 + Fn1 by FVSUM_1:71;
             G1 = G0^<* G1/.(len G1) *> by FINSEQ_5:21,A7
             .= G0^<* Gn1 *> by A7,A8,PARTFUN1:def 6; then
A15:         Sum G1 = Sum G0 + Gn1 by FVSUM_1:71;
             for i be Nat st i in dom F0 holds F0.i <= G0.i
             proof
               let i be Nat;
               assume
A16:           i in dom F0;
A17:           F0.i = (F1|(dom F0)).i by A9,FINSEQ_1:def 3
               .= F1.i by A16,FUNCT_1:49;
               G0.i = G1.i by A10,A16,FUNCT_1:49;
               hence thesis by A11,A16,A17,A7;
             end; then
A19:         Sum F0 <= Sum G0 by A6,A9;
             Fn1 <= Gn1 by A7,A8;
             hence thesis by A14,A15,A19,XREAL_1:7;
           end;
           hence thesis;
         end;
         hence thesis;
       end;
       for n being Nat holds P[n] from NAT_1:sch 2(A2,A5);
       hence Sum F <= Sum G by A1;
     end;
     hence thesis;
   end;
