
theorem Th15:
  for L being non empty multMagma for B being non empty AlgebraStr
over L for A being 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 addF of B)||the carrier of A).[x9,y9] by A1,FUNCT_1:49
    .= x9+ y9 by Def3;
end;
