reserve s for set,
  i,j for natural Number,
  k for Nat,
  x,x1,x2,x3 for Real,
  r,r1,r2,r3,r4 for Real,
  F,F1,F2,F3 for real-valued FinSequence,
  R,R1,R2 for Element of i-tuples_on REAL;

theorem Th93:
  for F being FinSequence of REAL holds Product F = multreal $$ F
proof
  set g = multreal, h = multcomplex;
  let F be FinSequence of REAL;
  rng F c= COMPLEX by NUMBERS:11,XBOOLE_1:1;
  then reconsider f = F as FinSequence of COMPLEX by FINSEQ_1:def 4;
  defpred P[Nat] means g $$ (finSeg $1,[#](F,the_unity_wrt g)) = h $$ (finSeg
  $1,[#](f,the_unity_wrt h));
  consider n being Nat such that
A1: dom f = Seg n by FINSEQ_1:def 2;
A2: g $$ F = g $$ (finSeg n,[#](F,the_unity_wrt g)) & h $$ f = h $$ (finSeg
  n, [#](f,the_unity_wrt h)) by A1,SETWOP_2:def 2;
A3: for k being Nat st P[k] holds P[k+1]
  proof
    set j = [#](f,the_unity_wrt h);
    set i = [#](F,the_unity_wrt g);
    let k be Nat;
    assume
A4: P[k];
A5: i.(k+1) = j.(k+1)
    proof
      per cases;
      suppose
A6:     k+1 in dom f;
        then j.(k+1) = F.(k+1) by SETWOP_2:20
          .= i.(k+1) by A6,SETWOP_2:20;
        hence thesis;
      end;
      suppose
A7:     not k+1 in dom f;
        then j.(k+1) = the_unity_wrt h by SETWOP_2:20
          .= i.(k+1) by A7,BINOP_2:6,7,SETWOP_2:20;
        hence thesis;
      end;
    end;
A8: not (k + 1) in Seg k by FINSEQ_3:8;
    reconsider k as Element of NAT by ORDINAL1:def 12;
    g $$ (finSeg (k+1),i)
       = g $$ (finSeg k \/ {.In(k+1,NAT).},i) by FINSEQ_1:9
      .= g.(g $$(finSeg k,i),i.(k+1)) by A8,SETWOP_2:2
      .= g $$(finSeg k,i) * (i.(k+1)) by BINOP_2:def 11
      .= h.(h $$(finSeg k,j),j.(k+1)) by A4,A5,BINOP_2:def 5
      .= h $$ (finSeg k \/ {.In(k+1,NAT).},j) by A8,SETWOP_2:2
      .= h $$ (finSeg (k+1),j) by FINSEQ_1:9;
    hence thesis;
  end;
A9: Seg 0 = {}.NAT;
  then g $$ (finSeg 0,[#](F,the_unity_wrt g)) = the_unity_wrt h by BINOP_2:6,7
,SETWISEO:31
    .= h $$ (finSeg 0,[#](f,the_unity_wrt h)) by A9,SETWISEO:31;
  then
A10: P[0];
  for k being Nat holds P[k] from NAT_1:sch 2(A10,A3);
  then g $$ F = h $$ f by A2;
  hence thesis by Def13;
end;
