let L be 1-sorted ; :: thesis: for A, B being AlgebraStr over L st A is Subalgebra of B & B is Subalgebra of A holds
AlgebraStr(# the carrier of A, the addF of A, the multF of A, the ZeroF of A, the OneF of A, the lmult of A #) = AlgebraStr(# the carrier of B, the addF of B, the multF of B, the ZeroF of B, the OneF of B, the lmult of B #)

let A, B be AlgebraStr over L; :: thesis: ( A is Subalgebra of B & B is Subalgebra of A implies AlgebraStr(# the carrier of A, the addF of A, the multF of A, the ZeroF of A, the OneF of A, the lmult of A #) = AlgebraStr(# the carrier of B, the addF of B, the multF of B, the ZeroF of B, the OneF of B, the lmult of B #) )
assume that
A1: A is Subalgebra of B and
A2: B is Subalgebra of A ; :: thesis: AlgebraStr(# the carrier of A, the addF of A, the multF of A, the ZeroF of A, the OneF of A, the lmult of A #) = AlgebraStr(# the carrier of B, the addF of B, the multF of B, the ZeroF of B, the OneF of B, the lmult of B #)
A3: the carrier of B c= the carrier of A by ;
A4: the carrier of A c= the carrier of B by ;
then A5: the carrier of A = the carrier of B by ;
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
.= the lmult of B by ;
A8: ( 0. A = 0. B & 1. A = 1. B ) by ;
A9: the multF of A = the multF of B || the carrier of A by
.= the multF of B by A5 ;
the addF of A = the addF of B || the carrier of A by
.= the addF of B by A5 ;
hence AlgebraStr(# the carrier of A, the addF of A, the multF of A, the ZeroF of A, the OneF of A, the lmult of A #) = AlgebraStr(# the carrier of B, the addF of B, the multF of B, the ZeroF of B, the OneF of B, the lmult of B #) by ; :: thesis: verum