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

theorem Th26:
  for A being non-empty MSAlgebra over MS st the carrier of MS = {
  0} holds MSAlg (1-Alg A) = the MSAlgebra of A
proof
  let A be non-empty MSAlgebra over MS;
  reconsider c = the_sort_of MSAlg (1-Alg A) as Component of the Sorts of
  MSAlg (1-Alg A) by MSUALG_1:def 12;
  assume the carrier of MS = {0};
  then MSSign 1-Alg A = the ManySortedSign of MS by Th23;
  then reconsider M1A = MSAlg (1-Alg A) as strict MSAlgebra over MS;
  reconsider M1A as non-empty strict MSAlgebra over MS by MSUALG_1:def 3;
A1: 1-Alg M1A = UAStr(#the_sort_of M1A, the_charact_of M1A#) by MSUALG_1:def 14
;
  reconsider c as Component of the Sorts of M1A;
A2: 1-Alg(MSAlg (1-Alg A)) = UAStr(#the_sort_of (MSAlg (1-Alg A)),
    the_charact_of (MSAlg (1-Alg A))#) by MSUALG_1:def 14;
  then
A3: the charact of 1-Alg A = the_charact_of MSAlg (1-Alg A) by MSUALG_1:9
    .= the Charact of M1A by MSUALG_1:def 13
    .= the charact of 1-Alg M1A by A1,MSUALG_1:def 13;
  c = the_sort_of M1A by MSUALG_1:def 12;
  then the carrier of 1-Alg A = the carrier of 1-Alg M1A by A2,A1,MSUALG_1:9;
  hence thesis by A3,Th24;
end;
