
theorem Th16:
  for L be non empty multMagma for B be non empty AlgebraStr over
  L for A be non empty Subalgebra of B holds for x,y being Element of B, x9,y9
  being Element of A st x = x9 & y = y9 holds x*y = x9* y9
proof
  let L be non empty multMagma;
  let B be non empty AlgebraStr over L;
  let A be non empty Subalgebra of B;
  let x,y be Element of B, x9,y9 be Element of A such that
A1: x = x9 & y = y9;
  [x9,y9] in [:the carrier of A,the carrier of A:] by ZFMISC_1:87;
  hence x*y = ((the multF of B)||the carrier of A).[x9,y9] by A1,FUNCT_1:49
    .= x9* y9 by Def3;
end;
