reserve L for non empty doubleLoopStr;
reserve a,b,c,x,y,z for Element of L;

theorem
  L is leftQuasi-Field iff (for a holds a + 0.L = a) & (for a ex x st a+
x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.
L <> 1.L & (for a holds a * 1.L = a) & (for a holds 1.L * a = a) & (for a,b st
a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x,y st a<>0.
L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a implies x=y)
& (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & for a,b,c holds a*(b+
  c) = a*b + a*c
proof
  thus L is leftQuasi-Field implies (for a holds a + 0.L = a) & (for a ex x st
a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) &
  0.L <> 1.L & (for a holds a * 1.L = a) & (for a holds 1.L * a = a) & (for a,b
st a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x,y st a
<>0.L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a implies x
=y) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & for a,b,c holds a
*(b+c) = a*b + a*c by ALGSTR_1:6,16,RLVECT_1:def 2,def 3,def 4,STRUCT_0:def 8
,VECTSP_1:def 2;
  assume that
A1: for a holds a + 0.L = a and
A2: for a ex x st a+x = 0.L and
A3: for a,b,c holds (a+b)+c = a+(b+c) and
A4: for a,b holds a+b = b+a and
A5: ( 0.L <> 1.L & for a holds a * 1.L = a )&( ( for a holds 1.L * a = a
  )& for a, b st a<>0.L ex x st a*x=b ) & ( ( for a,b st a<>0.L ex x st x*a=b)&
  for a,x,y st a<>0.L holds a*x=a*y implies x=y )&( ( for a,x,y st a<>0.L holds
x*a=y*a implies x=y)& for a holds a*0.L = 0.L ) &( ( for a holds 0.L*a = 0.L)&
  for a,b,c holds a*(b+c) = a*b + a*c);
A6: for a holds 0.L + a = a
  proof
    let a;
    thus 0.L + a = a + 0.L by A4
      .= a by A1;
  end;
A7: for a,b ex x st a+x=b
  proof
    let a,b;
    consider y such that
A8: a+y = 0.L by A2;
    take x = y+b;
    thus a+x = 0.L + b by A3,A8
      .= b by A6;
  end;
A9: for a,b ex x st x+a=b
  proof
    let a,b;
    consider x such that
A10: a+x=b by A7;
    take x;
    thus thesis by A4,A10;
  end;
A11: for a,x,y holds a+x=a+y implies x=y
  proof
    let a,x,y;
    consider z such that
A12: z+a = 0.L by A1,A2,A3,ALGSTR_1:3;
    assume a+x = a+y;
    then (z+a)+x = z+(a+y) by A3
      .= (z+a)+y by A3;
    hence x = 0.L + y by A6,A12
      .= y by A6;
  end;
  for a,x,y holds x+a=y+a implies x=y
  proof
    let a,x,y;
    assume x+a = y+a;
    then a+x= y+a by A4
      .= a+y by A4;
    hence thesis by A11;
  end;
  hence thesis by A1,A3,A4,A5,A6,A7,A9,A11,ALGSTR_1:6,16,def 2,RLVECT_1:def 2
,def 3,def 4,STRUCT_0:def 8,VECTSP_1:def 2,def 6;
end;
