
theorem
  for L being 1-sorted for A,B being AlgebraStr over L st
  A is Subalgebra of B & B is Subalgebra of A holds
  the AlgebraStr of A = the AlgebraStr of B
proof
  let L be 1-sorted;
  let A,B be AlgebraStr over L such that
A1: A is Subalgebra of B and
A2: B is Subalgebra of A;
A3: the carrier of B c= the carrier of A by A2,Def3;
A4: the carrier of A c= the carrier of B by A1,Def3;
  then
A5: the carrier of A = the carrier of B by A3,XBOOLE_0:def 10;
A6: dom (the lmult of B) = [:the carrier of L,the carrier of B:] by Th2;
A7: the lmult of A = (the lmult of B)|[:the carrier of L,the carrier of A:]
  by A1,Def3
    .= the lmult of B by A3,A6,RELAT_1:68,ZFMISC_1:96;
A8: 0.A = 0.B & 1.A = 1.B by A1,Def3;
A9: the multF of A = (the multF of B)||the carrier of A by A1,Def3
    .= the multF of B by A5;
  the addF of A = (the addF of B)||the carrier of A by A1,Def3
    .= the addF of B by A5;
  hence thesis by A4,A3,A9,A7,A8,XBOOLE_0:def 10;
end;
