 reserve j for set;
 reserve p,r for Real;
 reserve S,T,F for RealNormSpace;
 reserve x0 for Point of S;
 reserve g for PartFunc of S,T;
 reserve c for constant sequence of S;
 reserve R for RestFunc of S,T;
 reserve G for RealNormSpace-Sequence;
 reserve i for Element of dom G;
 reserve f for PartFunc of product G,F;
 reserve x for Element of product G;
reserve G for RealNormSpace-Sequence;
reserve F for RealNormSpace;
reserve i for Element of dom G;
reserve f,f1,f2 for PartFunc of product G, F;
reserve x for Point of product G;
reserve X for set;

theorem Th42:
for S be RealNormSpace, h,g be FinSequence of S
    st  len h = len g + 1 &
       (for i be Nat st i in dom g holds g/.i = h /.i - h/.(i+1)) holds
   h /.1 - h/.(len h) = Sum g
proof
   let S be RealNormSpace, h,g be FinSequence of S;
   assume that
A1: len h = len g + 1 and
A2: for i be Nat st i in dom g holds g/.i = h/.i - h/.(i+1);

   consider F be sequence of  the carrier of S such that
A3:  Sum g = F.(len g) & F.0 = 0.S
  & for j be Nat,v be Element of S
    st j < len g & v = g.(j + 1) holds F.(j + 1) = F.j + v
   by RLVECT_1:def 12;

   per cases;
   suppose len g = 0;
    hence thesis by A3,A1,RLVECT_1:15;
   end;
   suppose A4:len g > 0;
    defpred P[Nat] means $1 <= len g implies F.$1 = h/.1 - h/.($1+1);
A5: P[1]
    proof
     assume A6: 1 <= len g; then
     1 in Seg len g; then
A7:  1 in dom g by FINSEQ_1:def 3;
     reconsider zz0=0 as Element of NAT;
     g/.1 = g.( zz0 + 1 ) by A7,PARTFUN1:def 6; then
     F.(zz0 + 1) = F.0 + g/.1 by A3,A6
                .= g/.1 by A3,RLVECT_1:4;
     hence F.1 = h/.1 - h/.(1+1) by A7,A2;
    end;

A8: for j be Nat st 1 <= j holds P[j] implies P[j+1]
    proof
     let j be Nat;
     assume 1 <= j;
     assume A9: P[j];
     assume A10:j+1 <= len g; then
A12: j+1 in dom g by XREAL_1:38,FINSEQ_3:25; then
A13: g.(j+1)=g/.(j+1) by PARTFUN1:def 6;
     F.(j+1) = F.j + g/.(j+1) by A13,A10,A3,NAT_1:13
            .=F.j + (h/.(j+1) - h/.(j+1+1)) by A2,A12
            .= h/.1 - h/.(j+1) + h/.(j+1) - h/.(j+1+1)
                  by A9,A10,NAT_1:13,RLVECT_1:28
            .= h/.1 -(h/.(j+1) - h/.(j+1)) - h/.(j+1+1) by RLVECT_1:29
            .= h/.1 -0.S - h/.(j+1+1) by RLVECT_1:15;
     hence thesis by RLVECT_1:13;
    end;
A14:1 <= len g by A4,NAT_1:14;
    for i be Nat st 1 <= i holds P[i] from NAT_1:sch 8(A5,A8);
    hence thesis by A3,A1,A14;
   end;
end;
