
theorem Th38:
  for L be non empty ZeroStr for z0,z1 be Element of L holds <%z0,
z1%>.0 = z0 & <%z0,z1%>.1 = z1 & for n be Nat st n >= 2 holds <%z0,z1%>.n = 0.L
proof
  let L be non empty ZeroStr;
  let z0,z1 be Element of L;
  0 in NAT;
  then
A1: 0 in dom 0_.(L) by FUNCT_2:def 1;
  thus <%z0,z1%>.0 = (0_.(L)+*(0,z0)).0 by FUNCT_7:32
    .= z0 by A1,FUNCT_7:31;
  1 in NAT;
  then 1 in dom (0_.(L)+*(0,z0)) by FUNCT_2:def 1;
  hence <%z0,z1%>.1 = z1 by FUNCT_7:31;
  let n be Nat;
A2: n in NAT by ORDINAL1:def 12;
  assume
A3: n >= 2;
  then n >= 1+1;
  then n > 0+1 by NAT_1:13;
  hence <%z0,z1%>.n = (0_.(L)+*(0,z0)).n by FUNCT_7:32
    .= (0_.(L)).n by A3,FUNCT_7:32
    .= 0.L by A2,FUNCOP_1:7;
end;
