let A be non empty RelStr ; :: thesis: for a1, a2 being Element of A st A is connected & a1 <> a2 & not a1 <= a2 holds

a2 <= a1

let a1, a2 be Element of A; :: thesis: ( A is connected & a1 <> a2 & not a1 <= a2 implies a2 <= a1 )

assume that

A1: A is connected and

A2: a1 <> a2 ; :: thesis: ( a1 <= a2 or a2 <= a1 )

( [a1,a2] in the InternalRel of A or [a2,a1] in the InternalRel of A ) by A1, A2, RELAT_2:def 6;

hence ( a1 <= a2 or a2 <= a1 ) by ORDERS_2:def 5; :: thesis: verum

a2 <= a1

let a1, a2 be Element of A; :: thesis: ( A is connected & a1 <> a2 & not a1 <= a2 implies a2 <= a1 )

assume that

A1: A is connected and

A2: a1 <> a2 ; :: thesis: ( a1 <= a2 or a2 <= a1 )

( [a1,a2] in the InternalRel of A or [a2,a1] in the InternalRel of A ) by A1, A2, RELAT_2:def 6;

hence ( a1 <= a2 or a2 <= a1 ) by ORDERS_2:def 5; :: thesis: verum