let L be non empty multLoopStr ; :: thesis: for a being Element of L st ( for a being Element of L holds a * (1. L) = a ) & ( for a being Element of L ex x being Element of L st a * x = 1. L ) & ( for a, b, c being Element of L holds (a * b) * c = a * (b * c) ) holds

(1. L) * a = a * (1. L)

let a be Element of L; :: thesis: ( ( for a being Element of L holds a * (1. L) = a ) & ( for a being Element of L ex x being Element of L st a * x = 1. L ) & ( for a, b, c being Element of L holds (a * b) * c = a * (b * c) ) implies (1. L) * a = a * (1. L) )

assume that

A1: for a being Element of L holds a * (1. L) = a and

A2: for a being Element of L ex x being Element of L st a * x = 1. L and

A3: for a, b, c being Element of L holds (a * b) * c = a * (b * c) ; :: thesis: (1. L) * a = a * (1. L)

consider x being Element of L such that

A4: a * x = 1. L by A2;

thus (1. L) * a = a * (x * a) by A3, A4

.= a * (1. L) by A1, A2, A3, A4, Lm11 ; :: thesis: verum

(1. L) * a = a * (1. L)

let a be Element of L; :: thesis: ( ( for a being Element of L holds a * (1. L) = a ) & ( for a being Element of L ex x being Element of L st a * x = 1. L ) & ( for a, b, c being Element of L holds (a * b) * c = a * (b * c) ) implies (1. L) * a = a * (1. L) )

assume that

A1: for a being Element of L holds a * (1. L) = a and

A2: for a being Element of L ex x being Element of L st a * x = 1. L and

A3: for a, b, c being Element of L holds (a * b) * c = a * (b * c) ; :: thesis: (1. L) * a = a * (1. L)

consider x being Element of L such that

A4: a * x = 1. L by A2;

thus (1. L) * a = a * (x * a) by A3, A4

.= a * (1. L) by A1, A2, A3, A4, Lm11 ; :: thesis: verum