
theorem lemred1:
for F being Field
for b1,c1,b2,c2 being Element of F
holds <%c1,b1%> *' <%c2,b2%> = <%c1*c2,b1*c2+b2*c1,b1*b2%>
proof
let F be Field; let b1,c1,b2,c2 be Element of F;
set p1 = <%c1,b1%>, p2 = <%c2,b2%>, q = <%c1*c2,b1*c2+b2*c1,b1*b2%>;
J: 4 -' 1 = 4 - 1 by XREAL_0:def 2; then
K: 2 -' 1 = 2 - 1 & 3 -' 1 = 3 - 1 & 3 -' 2 = 3 - 2 & 4 -' 1 = 3
   by XREAL_0:def 2;
A: dom(p1*'p2) = NAT by FUNCT_2:def 1 .= dom q by FUNCT_2:def 1;
now let o be object;
  assume o in dom q;
  then reconsider i = o as Element of NAT;
  consider r being FinSequence of the carrier of F such that
  B1: len r = i+1 & (p1*'p2).i = Sum r &
      for k being Element of NAT st k in dom r
      holds r.k = p1.(k-'1) * p2.(i+1-'k) by POLYNOM3:def 9;
  i <= 2 implies i = 0 or ... or i = 2; then
  per cases;
  suppose C: i = 0;
    then B2: r = <*r.1*> by B1,FINSEQ_1:40;
    then dom r = {1} by FINSEQ_1:2,FINSEQ_1:38;
    then 1 in dom r by TARSKI:def 1;
    then r.1 = p1.(1-'1) * p2.(0+1-'1) by C,B1
            .= p1.(1-'1) * p2.0 by NAT_2:8
            .= p1.0 * p2.0 by NAT_2:8
            .= c1 * p2.0 by POLYNOM5:38
            .= c1 * c2 by POLYNOM5:38;
    then Sum r = c1*c2 by B2,RLVECT_1:44;
    hence (p1*'p2).o = q.o by B1,C,qua1;
    end;
  suppose C: i = 1;
    then B3: r = <*r.1,r.2*> by B1,FINSEQ_1:44;
    B4: dom r = {1,2} by B1,C,FINSEQ_1:def 3,FINSEQ_1:2;
    then 1 in dom r by TARSKI:def 2;
    then B5: r.1 = p1.(1-'1) * p2.(1+1-'1) by C,B1
                .= p1.0 * p2.1 by K,NAT_2:8
                .= c1 * p2.1 by POLYNOM5:38
                .= c1 * b2 by POLYNOM5:38;
    2 in dom r by B4,TARSKI:def 2;
    then r.2 = p1.(2-'1) * p2.(1+1-'2) by C,B1
            .= p1.1 * p2.0 by K,NAT_2:8
            .= b1 * p2.0 by POLYNOM5:38
            .= b1 * c2 by POLYNOM5:38;
    then Sum r = c1 * b2 + b1 * c2 by B3,B5,RLVECT_1:45
              .= b1 * c2 + b2 * c1 by GROUP_1:def 12;
    hence (p1*'p2).o = q.o by B1,C,qua1;
    end;
  suppose C: i = 2;
    then B3: r = <*r.1,r.2,r.3*> by B1,FINSEQ_1:45;
    B4: dom r = Seg 3 by B1,C,FINSEQ_1:def 3
             .= Seg 2 \/ {2+1} by FINSEQ_1:9
             .= {1,2,3} by FINSEQ_1:2,ENUMSET1:3;
    then 1 in dom r by ENUMSET1:def 1;
    then B5: r.1 = p1.(1-'1) * p2.(2+1-'1) by C,B1
                .= p1.0 * p2.2 by K,NAT_2:8
                .= p1.0 * 0.F by POLYNOM5:38;
    2 in dom r by B4,ENUMSET1:def 1;
    then B6: r.2 = p1.1 * p2.1 by K,C,B1
                .= b1 * p2.1 by POLYNOM5:38
                .= b1 * b2 by POLYNOM5:38;
    3 in dom r by B4,ENUMSET1:def 1;
    then r.3 = p1.2 * p2.(2+1-'3) by K,C,B1
            .= 0.F * p2.(2+1-'3) by POLYNOM5:38;
    then Sum r = 0.F + b1 * b2 + 0.F by B3,B5,B6,RLVECT_1:46;
    hence (p1*'p2).o = q.o by B1,C,qua1;
    end;
  suppose C: i > 2;
    D: len q <= 3 by qua2;
    C1: i >= 2 + 1 by C,NAT_1:13;
    then E: q.i = 0.F by D,XXREAL_0:2,ALGSEQ_1:8;
    len p1 <= 2 & len p2 <= 2 by POLYNOM5:39;
    then len p1 + len p2 <= 2 + 2 by XREAL_1:7;
    then F: len p1 + len p2 -' 1 <= 2 + 2 -' 1 by lemmonus;
    len(p1*'p2) <= len p1 + len p2 -' 1 by leng;
    then len(p1*'p2) <= 2 + 2 -' 1 by F,XXREAL_0:2;
    hence (p1*'p2).o = q.o by J,C1,XXREAL_0:2,E,ALGSEQ_1:8;
    end;
  end;
hence thesis by A;
end;
