
theorem Th1:
for V being RealLinearSpace,
    A being Subset of V,
    x be FinSequence of V,
    r be FinSequence of REAL
     st rng x c= A & len x = len r
   holds Sum (r(*)x) in Lin A
proof
let V be RealLinearSpace,
    A be Subset of V;
defpred P[Nat] means
 for x be FinSequence of V,
    r be FinSequence of REAL
     st $1 = len x & rng x c= A & len x = len r
   holds Sum (r(*)x) in Lin A;
A1:P[0]
proof
let x be FinSequence of V,
    r be FinSequence of REAL;
   assume A2: 0 = len x & rng x c= A & len x = len r;
set y = r(*)x;
len y = len x by DefR;
then y = <*> (the carrier of V) by A2; then
Sum y = 0.V by RLVECT_1:43; then
Sum y = 0.(Lin A) by RLSUB_1:11;
hence Sum y in Lin A by STRUCT_0:def 5;
end;
A3: for k be Nat st P[k] holds P[k+1]
proof
  let k be Nat;
  assume
A4: P[k];
  let x be FinSequence of V,
    r be FinSequence of REAL;
    assume
A5: k+1 = len x & rng x c= A & len x = len r;
set y = r(*)x;
set x1=x | k;
set y1=y | k;
set r1=r | k;
A6:k <= k+1 by NAT_1:11;
AA: len y = len x by DefR; then
A7: len x1 = k & len y1 = k & len r1 = k by A5,FINSEQ_1:59,NAT_1:11;
rng x1 c= rng x by RELAT_1:70; then
A8: k= len x1 & rng x1 c= A & len x1=len y1
  & len x1 = len r1 by A7,A5;
D1:for i be Nat st 1<=i & i <= len x1
 holds y1.i = r1/.i*x1/.i
proof
let i be Nat;
   assume A10: 1<=i & i <= len x1; then
   A11: i <= len x by A6,A7,A5,XXREAL_0:2; then
   A13: i in dom x by A10,FINSEQ_3:25;
   A14: i in dom r by A10,A11,A5,FINSEQ_3:25;
   A16: i in dom x1 by A10,FINSEQ_3:25;
   A19: y1.i =y.i by FUNCT_1:49,A10,FINSEQ_1:1,A7;
   i in Seg k by A10,A7; then
   i in dom r1 by FINSEQ_1:def 3,A7; then
   A20: r1/.i = r1.i by PARTFUN1:def 6
             .= r.i by FUNCT_1:49,A10,FINSEQ_1:1,A7
             .= r/.i by A14,PARTFUN1:def 6;
   x1/.i = x1.i by A16,PARTFUN1:def 6
             .= x.i by FUNCT_1:49,A10,FINSEQ_1:1,A7
             .= x/.i by A13,PARTFUN1:def 6;
   hence thesis by A10,A19,A20,A11,DefR;
end;
  y1 = r1(*)x1 by A7,D1,DefR; then
A22: Sum y1 in Lin A by A4,A8;
  1 <= k+1 by NAT_1:11; then
A24: y.(len y) = r/.(k+1)*x/.(k+1) by A5,AA,DefR;
  dom y1 = Seg k by A7,FINSEQ_1:def 3; then
A26: Sum y = Sum y1 + r/.(k+1)*x/.(k+1) by RLVECT_1:38,A7,A5,DefR,A24;
A27:dom x = Seg (k+1) by A5,FINSEQ_1:def 3; then
A29:x/.(k+1) = x.(k+1) by PARTFUN1:def 6,FINSEQ_1:4;
  k+1 in Seg (k+1) by FINSEQ_1:4; then
  x/.(k+1) in rng x by A29,A27,FUNCT_1:def 3; then
  r/.(k+1)*x/.(k+1) in Lin A by RLSUB_1:21,RLVECT_3:15,A5;
  hence Sum y in Lin A by A26,A22,RLSUB_1:20;
end;
for k be Nat holds P[k] from NAT_1:sch 2(A1,A3);
hence thesis;
end;
