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