
theorem Th28:
  for L being non empty multLoopStr_0 holds 1_.(L) = <%1.L%>
proof
  let L be non empty multLoopStr_0;
A1: dom 0_.(L) = NAT by FUNCT_2:def 1;
  now
    let x be object;
    assume x in NAT;
    then reconsider n = x as Element of NAT;
    per cases;
    suppose
A2:   x = 0;
      hence (1_.(L)).x = 1.L by A1,FUNCT_7:31
        .= <%1.L%>.x by A2,ALGSEQ_1:def 5;
    end;
    suppose
A3:   n <> 0;
      then
A4:   n = 1 or n > 1 by NAT_1:53;
      thus (1_.(L)).x = (0_.(L)).n by A3,FUNCT_7:32
        .= 0.L by FUNCOP_1:7
        .= <%1.L%>.x by A4,POLYNOM5:32;
    end;
  end;
  hence thesis by FUNCT_2:12;
end;
