reserve x,y,X,Y for set,
  k,l,n for Nat,
  i,i1,i2,i3,j for Integer,
  G for Group,
  a,b,c,d for Element of G,
  A,B,C for Subset of G,
  H,H1,H2, H3 for Subgroup of G,
  h for Element of H,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem
  Product(F - {1_G}) = Product(F)
proof
  defpred P[FinSequence of the carrier of G] means Product($1 - {1_G}) =
  Product$1;
A1: for F,a st P[F] holds P[F ^ <* a *>]
  proof
    let F,a;
    assume
A2: P[F];
A3: Product((F ^ <* a *>) - {1_G}) = Product(((F - {1_G}) ^ (<* a *> - {
    1_G}))) by FINSEQ_3:73
      .= Product F * Product(<* a *> - {1_G}) by A2,FINSOP_1:5;
    now
      per cases;
      suppose
A4:     a = 1_G;
        then a in {1_G} by TARSKI:def 1;
        then <* a *> - {1_G} = <*> the carrier of G by FINSEQ_3:76;
        then Product(<* a *> - {1_G}) = 1_G by Th8;
        hence thesis by A3,A4,FINSOP_1:4;
      end;
      suppose
        a <> 1_G;
        then not a in {1_G} by TARSKI:def 1;
        then <* a *> - {1_G} = <* a *> by FINSEQ_3:75;
        hence thesis by A3,FINSOP_1:5;
      end;
    end;
    hence thesis;
  end;
A5: P[<*> the carrier of G] by FINSEQ_3:74;
  for F holds P[F] from FINSEQ_2:sch 2(A5,A1);
  hence thesis;
end;
