reserve FS for non empty doubleLoopStr;
reserve F for Field;
reserve R for Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr,
  x, y, z for Scalar of R;
reserve SF for Skew-Field,
  x, y, z for Scalar of SF;

theorem Th8:
  for SF being non degenerated almost_left_invertible associative
  Abelian add-associative right_zeroed right_complementable well-unital
distributive non empty doubleLoopStr, x,y,z being Element of SF holds x * y =
  x * z & x<>0.SF implies y = z
proof
  let SF be non degenerated almost_left_invertible associative Abelian
add-associative right_zeroed right_complementable well-unital distributive non
  empty doubleLoopStr, x,y,z be Element of SF;
  assume that
A1: x * y = x * z and
A2: x<>0.SF;
  consider u being Element of SF such that
A3: x * u = 1_SF by A2,Th6;
A4: u * x = 1_SF by A3,Th7;
  then y = (u*x)*y
    .= u*(x*z) by A1,GROUP_1:def 3
    .= 1_SF*z by A4,GROUP_1:def 3
    .= z;
  hence thesis;
end;
