
theorem Th4:
for V being RealUnitarySpace,
    u be Point of V,
    x be FinSequence of V
holds u .|. (Sum x) = Sum (u .|. x)
proof
let V be RealUnitarySpace,
    u be Point of V,
    x be FinSequence of V;
defpred P[Nat] means
 for x be FinSequence of V st $1 = len x holds u .|. (Sum x) = Sum (u .|. x);
A1:P[0]
 proof
 let x be FinSequence of V;
  assume A2:0=len x; then
  x = <*> the carrier of V; then
A3: Sum x =0.V by RLVECT_1:43;
set r = u .|. x;
len r = len x by DefSK;
then r = <*> REAL by A2;
hence u .|. (Sum x) = Sum r by A3,BHSP_1:15,RVSUM_1:72;
end;

A4:for k be Nat st P[k] holds P[k+1]
 proof
   let k be Nat;
   assume
A5: P[k];
   let x be FinSequence of V;
   set r = u .|. x;
   set x1=x | k;
   set r1=r | k;
   assume
A6: k+1=len x; then
A7: 1 <= k+1 & k+1 <= len x by NAT_1:11;
A8: k <= k+1 by NAT_1:11;
B1: len r = len x by DefSK; then
A9: len x1 = k & len r1 = k by A6,FINSEQ_1:59,NAT_1:11;
for i be Nat st 1<=i & i <= len x1 holds r1.i = u .|. (x1/.i)
proof
let i be Nat;
   assume A10: 1<=i & i <= len x1; then
   A11: i <= len x by A8,A9,A6,XXREAL_0:2; then
   A13: i in dom x by A10,FINSEQ_3:25;
   A20: r1.i = r.i by FUNCT_1:49,A9,A10,FINSEQ_1:1;
   i in dom x1 by A10,FINSEQ_3:25; then
   x1/.i = x1.i by PARTFUN1:def 6
        .= x.i by FUNCT_1:49,A9,A10,FINSEQ_1:1
        .= x/.i by A13,PARTFUN1:def 6;
   hence thesis by A10,A20,A11,DefSK;
end; then
A22b: r1 = u .|. x1 by A9,DefSK;
     A23a:r= r1 ^ <*(r . (k + 1))*> by B1,FINSEQ_3:55,A6
           .=r1 ^ <*u .|. (x/.(k+1)) *> by A7,DefSK;
     A24: len x = len x1 + 1 by A6,FINSEQ_1:59,NAT_1:11;
     A25: dom x1 = Seg k by A9,FINSEQ_1:def 3;
     k+1 in Seg (k+1) by FINSEQ_1:4; then
     k+1 in dom x by A6,FINSEQ_1:def 3; then
     x/.(k+1) = x.(len x) by A6,PARTFUN1:def 6;
     hence u .|. (Sum x)
            = u .|. (Sum x1 + x/.(k+1)) by A24,A25,RLVECT_1:38
            .= u .|. (Sum x1) + u .|.(x/.(k+1)) by BHSP_1:2
            .= Sum r1  + u .|.(x/.(k+1)) by A22b,A5,A9
            .= Sum r by A23a,RVSUM_1:74;
    end;
B3: len (u .|. x) = len x by DefSK;
for k be Nat holds P[k] from NAT_1:sch 2(A1,A4);
hence thesis by B3;
end;
