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

theorem Th23:
  for h be Function of R,R, s be FinSequence of the carrier of R
   st h is additive holds h.(Sum s) = Sum(h*s)
   proof
     let h be Function of R,R, s be FinSequence of the carrier of R;
     assume
A1:  h is additive;
     defpred P[Nat] means
     for h being Function of R,R, s being FinSequence of R holds
     len s = $1 & h is additive implies h.(Sum s) = Sum (h*s);
A2:  P[0]
     proof
       let h be Function of R,R;
       let s be FinSequence of R;
       assume that
A3:    len s = 0 and
A4:    h is additive;
       Sum s = 0.R by A3,RLVECT_1:75; then
A5:    h.(Sum s) = 0.R by A4,RING_2:6;
       dom h = the carrier of R by FUNCT_2:def 1; then
       rng s c= dom h; then
       dom(h*s) = dom s by RELAT_1:27 .= Seg len s by FINSEQ_1:def 3; then
       h*s = <*>the carrier of R by A3;
       hence thesis by A5,RLVECT_1:43;
     end;
A6:  for n be Nat st P[n] holds P[n+1]
     proof
       let n;
       assume
A7:    P[n];
       let h be Function of R,R;
       let s be FinSequence of R;
       assume that
A8:    len s = n+1 and
A9:    h is additive;
       set s0=s|n;
       dom s = Seg(n+1) by A8,FINSEQ_1:def 3; then
       s.(n+1) in rng s by FUNCT_1:3,FINSEQ_1:4; then
       reconsider v=s.(n+1) as Element of R;
A10:   n = len s0 by A8,FINSEQ_1:59,NAT_1:11;
       1 <= n+1 <= len s by A8,NAT_1:11; then
A11:   s/.len s = s.(n+1) by A8,FINSEQ_4:15; then
A12:   s = (s|n)^<*v*> by A8,FINSEQ_5:21;
A13:   h*s = h*((s|n)^<*v*>) by A11,A8,FINSEQ_5:21
         .= (h*s0)^<*h.v*> by FINSEQOP:8;
       h.(Sum s) = h.(Sum s0 + Sum<*v*>) by A12,RLVECT_1:41
         .= h.(Sum s0 + v) by RLVECT_1:44
         .= h.(Sum s0) + h.v by A9
         .= Sum(h*s0) + h.v by A7,A9,A10
         .= Sum(h*s0) + Sum <*h.v*> by RLVECT_1:44
         .= Sum(h*s) by A13,RLVECT_1:41;
       hence thesis;
     end;
A14: for n be Nat holds P[n] from NAT_1:sch 2(A2,A6);
     len s is Nat;
     hence thesis by A1,A14;
   end;
