
theorem
for L being non empty multLoopStr_0
holds even_part 1_.(L) = 1_.(L) & odd_part 1_.(L) = 0_.(L)
proof
let L be non empty multLoopStr_0;
set e = even_part(1_.(L)), p = 1_.(L);
A1: dom p = NAT by FUNCT_2:def 1 .= dom e by FUNCT_2:def 1;
now let x be object;
  assume x in dom p;
  then reconsider i = x as Element of NAT by FUNCT_2:def 1;
  now per cases;
  case i is even;
    hence e.i = p.i by Def1;
    end;
  case A2: i is odd;
    hence e.i = 0.L by Def1 .= p.i by A2,POLYNOM3:30;
    end;
  end;
  hence p.x = e.x;
  end;
hence even_part 1_.(L) = 1_.(L) by A1,FUNCT_1:2;
set o = odd_part(1_.(L)), p = 0_.(L);
A3: dom p = NAT by FUNCT_2:def 1 .= dom o by FUNCT_2:def 1;
now let x be object;
  assume x in dom p;
  then reconsider i = x as Element of NAT by FUNCT_2:def 1;
  now per cases;
  case A4: i is odd;
    hence o.i = (1_.(L)).i by Def2
             .= 0.L by A4,POLYNOM3:30 .= p.i by FUNCOP_1:7;
    end;
  case i is even;
    hence o.i = 0.L by Def2 .= p.i by FUNCOP_1:7;
    end;
  end;
  hence p.x = o.x;
  end;
hence odd_part 1_.(L) = 0_.(L) by A3,FUNCT_1:2;
end;
