reserve a for set,
  i for Nat;
reserve MS for segmental non void 1-element ManySortedSign,
  A for non-empty MSAlgebra over MS;

theorem Th23:
  for A being non-empty MSAlgebra over MS st the carrier of MS = {
  0} holds MSSign 1-Alg A = the ManySortedSign of MS
proof
  let A be non-empty MSAlgebra over MS;
  reconsider ff1 = (*-->0)*(signature 1-Alg A) as Function of dom signature
  1-Alg A, {0}* by MSUALG_1:2;
A1: MSSign 1-Alg A = ManySortedSign (#{0},dom signature(1-Alg A),ff1,dom
    signature(1-Alg A)-->z#) by MSUALG_1:10;
  dom (the ResultSort of MS) = the carrier' of MS by FUNCT_2:def 1;
  then
A2: rng (the ResultSort of MS) <> {} by RELAT_1:42;
  consider k be Nat such that
A3: the carrier' of MS = Seg k by MSUALG_1:def 7;
A4: 1-Alg A = UAStr(#the_sort_of A, the_charact_of A#) by MSUALG_1:def 14;
A5: len signature (1-Alg A) = len the charact of 1-Alg A by UNIALG_1:def 4;
  then
A6: dom signature (1-Alg A) = dom (the charact of 1-Alg A) by FINSEQ_3:29
    .= dom the Charact of A by A4,MSUALG_1:def 13
    .= the carrier' of MS by PARTFUN1:def 2;
  then
A7: the carrier' of MS = dom ff1 by FUNCT_2:def 1;
  assume
A8: the carrier of MS = {0};
A9: for x being object st x in the carrier' of MS holds ((*-->0)*(signature (
  1-Alg A))).x = (the Arity of MS).x
  proof
    let x be object;
    assume x in the carrier' of MS;
    then reconsider x as OperSymbol of MS;
    x in Seg k by A3;
    then reconsider n = x as Nat;
    n in dom(signature (1-Alg A)) by A6;
    then
A10: n in dom the charact of 1-Alg A by A5,FINSEQ_3:29;
    reconsider
    h = (the charact of 1-Alg A).n as PartFunc of (the carrier of
    1-Alg A)*,the carrier of 1-Alg A;
    reconsider h as homogeneous quasi_total non empty PartFunc of (the carrier
    of 1-Alg A)*,the carrier of 1-Alg A by A10,MARGREL1:def 24;
    set aa = the Element of dom h;
    set ah = arity h;
    dom h c= (the carrier of 1-Alg A)* by RELAT_1:def 18;
    then aa in (the carrier of 1-Alg A)*;
    then reconsider bb = aa as FinSequence of the carrier of 1-Alg A by
FINSEQ_1:def 11;
A11: bb in dom h;
    h = (the Charact of A).x by A4,MSUALG_1:def 13
      .= Den(x,A) by MSUALG_1:def 6;
    then bb in Args(x,A) by A11,FUNCT_2:def 1;
    then bb in (len the_arity_of x)-tuples_on the_sort_of A by MSUALG_1:6;
    then
A12: len the_arity_of x = len bb by CARD_1:def 7
      .= ah by MARGREL1:def 25;
    ((*-->0)*(signature (1-Alg A))).x = (*-->0).((signature (1-Alg A)).x)
    by A6,FUNCT_1:13
      .= (*-->0).ah by A6,UNIALG_1:def 4
      .= ah |-> 0 by FINSEQ_2:def 6
      .= the_arity_of x by A8,A12,Th5
      .= (the Arity of MS).x by MSUALG_1:def 1;
    hence thesis;
  end;
  rng (the ResultSort of MS) c= {0} by A8,RELAT_1:def 19;
  then the carrier' of MS = dom (the ResultSort of MS) & rng (the ResultSort
  of MS) = {0} by A2,FUNCT_2:def 1,ZFMISC_1:33;
  then the carrier' of MS = dom the Arity of MS & the ResultSort of MSSign
  1-Alg A = the ResultSort of MS by A1,A6,FUNCOP_1:9,FUNCT_2:def 1;
  hence thesis by A8,A1,A7,A9,FUNCT_1:2;
end;
