
theorem Th17:
  for L be non empty multMagma for B be non empty AlgebraStr over
L for A be non empty Subalgebra of B holds for a being Element of L for x being
  Element of B, x9 being Element of A st x = x9 holds a * x = a * x9
proof
  let L be non empty multMagma;
  let B be non empty AlgebraStr over L;
  let A be non empty Subalgebra of B;
  let a be Element of L;
  let x be Element of B, x9 be Element of A such that
A1: x = x9;
  [a,x9] in [:the carrier of L,the carrier of A:] by ZFMISC_1:87;
  hence
  a * x = ((the lmult of B)|[:the carrier of L,the carrier of A:]).[a,x9]
  by A1,FUNCT_1:49
    .= a * x9 by Def3;
end;
