
theorem Th35:
  for L be add-associative right_zeroed right_complementable
  distributive commutative associative well-unital domRing-like non empty
  doubleLoopStr for x,y be Element of L holds <%x%>*'<%y%> = <%x*y%>
proof
  let L be add-associative right_zeroed right_complementable distributive
  commutative associative well-unital domRing-like non empty doubleLoopStr;
  let x,y be Element of L;
A1: len <%x%> <= 1 by ALGSEQ_1:def 5;
A2: len <%y%> <= 1 by ALGSEQ_1:def 5;
  per cases;
  suppose
A3: len <%x%> <> 0 & len <%y%> <> 0;
    x <> 0.L & y <> 0.L
    proof
      assume x = 0.L or y = 0.L;
      then <%x%> = 0_.(L) or <%y%> = 0_.(L) by Th34;
      hence contradiction by A3,POLYNOM4:3;
    end;
    then x*y <> 0.L by VECTSP_2:def 1;
    then
A4: len <%x*y%> = 1 by Th33;
    consider r be FinSequence of the carrier of L such that
A5: len r = 0+1 and
A6: (<%x%>*'<%y%>).0 = Sum r and
A7: for k be Element of NAT st k in dom r holds r.k = <%x%>.(k-'1) *
    <%y%>.(0+1-'k) by POLYNOM3:def 9;
    1 in dom r by A5,FINSEQ_3:25;
    then r.1 = <%x%>.(1-'1) * <%y%>.(0+1-'1) by A7
      .= <%x%>.0 * <%y%>.(1-'1) by XREAL_1:232
      .= <%x%>.0 * <%y%>.0 by XREAL_1:232
      .= <%x%>.0 * y by ALGSEQ_1:def 5
      .= x*y by ALGSEQ_1:def 5;
    then
A8: r = <*x*y*> by A5,FINSEQ_1:40;
A9: now
      let n be Nat;
      assume n < 1;
      then n < 0+1;
      then
A10:  n = 0 by NAT_1:13;
      hence (<%x%>*'<%y%>).n = x*y by A6,A8,RLVECT_1:44
        .= <%x*y%>.n by A10,ALGSEQ_1:def 5;
    end;
    <%x%>.(len <%x%> -'1) <> 0.L & <%y%>.(len <%y%> -'1) <> 0.L by A3,Lm2;
    then
A11: <%x%>.(len <%x%> -'1)*<%y%>.(len <%y%> -'1)<>0.L by VECTSP_2:def 1;
    len <%y%> >= 0+1 by A3,NAT_1:13;
    then
A12: len <%y%> = 1 by A2,XXREAL_0:1;
    len <%x%> >= 0+1 by A3,NAT_1:13;
    then len <%x%> = 1 by A1,XXREAL_0:1;
    then len (<%x%>*'<%y%>) = 1+1-1 by A12,A11,POLYNOM4:10;
    hence thesis by A9,A4,ALGSEQ_1:12;
  end;
  suppose
A13: len <%x%> = 0;
    then
A14: x=0.L by Th33;
    <%x%> = 0_.(L) by A13,POLYNOM4:5;
    hence <%x%>*'<%y%> = 0_.(L) by POLYNOM4:2
      .= <%0.L%> by Th34
      .= <%x*y%> by A14;
  end;
  suppose
A15: len <%y%> = 0;
    then
A16: y=0.L by Th33;
    <%y%> = 0_.(L) by A15,POLYNOM4:5;
    hence <%x%>*'<%y%> = 0_.(L) by POLYNOM3:34
      .= <%0.L%> by Th34
      .= <%x*y%> by A16;
  end;
end;
