reserve L for non empty addLoopStr;
reserve a,b,c,x for Element of L;
reserve L for non empty multLoopStr;
reserve a,b,c,x,y,z for Element of L;
reserve G for multGroup;
reserve a,b,c,x for Element of G;

theorem Th14:
  for L being non empty multLoopStr_0 holds L is multLoop_0-like
iff (for a,b be Element of L st a<>0.L ex x be Element of L st a*x=b) & (for a,
  b be Element of L st a<>0.L ex x be Element of L st x*a=b) & (for a,x,y be
Element of L st a<>0.L holds a*x=a*y implies x=y) & (for a,x,y be Element of L
st a<>0.L holds x*a=y*a implies x=y) & (for a be Element of L holds a*0.L = 0.L
  ) & for a be Element of L holds 0.L*a = 0.L
proof
  let L be non empty multLoopStr_0;
  hereby
    assume
A1: L is multLoop_0-like;
    then
A2: L is almost_invertible almost_cancelable;
    hence (for a,b be Element of L st a<>0.L ex x be Element of L st a*x=b) &
    for a,b be Element of L st a<>0.L ex x be Element of L st x*a=b;
    thus for a,x,y be Element of L st a<>0.L holds a*x=a*y implies x=y
         by A2,ALGSTR_0:def 20,def 36;
    thus for a,x,y be Element of L st a<>0.L holds x*a=y*a implies x=y
         by A2,ALGSTR_0:def 21,def 37;
    thus (for a be Element of L holds a*0.L = 0.L) & for a be Element of L
    holds 0.L*a = 0.L by A1;
  end;
  assume that
A3: ( for a,b be Element of L st a<>0.L ex x be Element of L st a*x=b)&
  for a,b be Element of L st a<>0.L ex x be Element of L st x*a=b and
A4: for a,x,y be Element of L st a<>0.L holds a*x=a*y implies x=y and
A5: for a,x,y be Element of L st a<>0.L holds x*a=y*a implies x=y and
A6: ( for a be Element of L holds a*0.L = 0.L)& for a be Element of L
  holds 0.L*a = 0.L;
A7: L is almost_right_cancelable
  proof
    let x being Element of L;
    assume
A8: x <> 0.L;
    let y,z be Element of L;
    assume y*x = z*x;
    hence thesis by A5,A8;
  end;
  L is almost_left_cancelable
  proof
    let x being Element of L;
    assume
A9: x <> 0.L;
    let y,z be Element of L;
    assume x*y = x*z;
    hence thesis by A4,A9;
  end;
  then L is almost_invertible almost_cancelable by A3,A7;
  hence thesis by A6;
end;
