
theorem
  for L being 1-sorted for A,B,C being AlgebraStr over L st
  A is Subalgebra of B & B is Subalgebra of C holds A is Subalgebra of C
proof
  let L be 1-sorted;
  let A,B,C be AlgebraStr over L such that
A1: A is Subalgebra of B and
A2: B is Subalgebra of C;
A3: the carrier of A c= the carrier of B by A1,Def3;
  then
A4: [:the carrier of A,the carrier of A:] c= [:the carrier of B,the carrier
  of B:] by ZFMISC_1:96;
  the carrier of B c= the carrier of C by A2,Def3;
  hence the carrier of A c= the carrier of C by A3;
  thus 1.A = 1.B by A1,Def3
    .= 1.C by A2,Def3;
  thus 0.A = 0.B by A1,Def3
    .= 0.C by A2,Def3;
  thus the addF of A = (the addF of B)||the carrier of A by A1,Def3
    .= ((the addF of C)||the carrier of B)||the carrier of A by A2,Def3
    .= (the addF of C)||the carrier of A by A4,FUNCT_1:51;
  thus the multF of A = (the multF of B)||the carrier of A by A1,Def3
    .= ((the multF of C)||the carrier of B)||the carrier of A by A2,Def3
    .= (the multF of C)||the carrier of A by A4,FUNCT_1:51;
A5: [:the carrier of L,the carrier of A:] c= [:the carrier of L,the carrier
  of B:] by A3,ZFMISC_1:96;
  thus the lmult of A = (the lmult of B)|[:the carrier of L,the carrier of A
  :] by A1,Def3
    .= ((the lmult of C)|[:the carrier of L,the carrier of B:])| [:the
  carrier of L,the carrier of A:] by A2,Def3
    .= (the lmult of C)|[:the carrier of L,the carrier of A:] by A5,FUNCT_1:51;
end;
