let S be OAffinSpace; :: thesis: for AS being non empty AffinStruct st AS = Lambda S holds
for a, b, c, d being Element of S
for a', b', c', d' being Element of AS st a = a' & b = b' & c = c' & d = d' holds
( a',b' // c',d' iff a,b '||' c,d )
let AS be non empty AffinStruct ; :: thesis: ( AS = Lambda S implies for a, b, c, d being Element of S
for a', b', c', d' being Element of AS st a = a' & b = b' & c = c' & d = d' holds
( a',b' // c',d' iff a,b '||' c,d ) )
assume A1: AS = Lambda S ; :: thesis: for a, b, c, d being Element of S
for a', b', c', d' being Element of AS st a = a' & b = b' & c = c' & d = d' holds
( a',b' // c',d' iff a,b '||' c,d )
let a, b, c, d be Element of S; :: thesis: for a', b', c', d' being Element of AS st a = a' & b = b' & c = c' & d = d' holds
( a',b' // c',d' iff a,b '||' c,d )
let a', b', c', d' be Element of AS; :: thesis: ( a = a' & b = b' & c = c' & d = d' implies ( a',b' // c',d' iff a,b '||' c,d ) )
assume A2: ( a = a' & b = b' & c = c' & d = d' ) ; :: thesis: ( a',b' // c',d' iff a,b '||' c,d )
thus ( a',b' // c',d' implies a,b '||' c,d ) :: thesis: ( a,b '||' c,d implies a',b' // c',d' )

proof

assume A3: [[a',b'],[c',d']] in the CONGR of AS ; :: according to ANALOAF:def 2 :: thesis: a,b '||' c,d
assume not [[a,b],[c,d]] in the CONGR of S ; :: according to ANALOAF:def 2,DIRAF:def 4 :: thesis: a,b // d,c
hence [[a,b],[d,c]] in the CONGR of S by A1, A2, A3, Def1; :: according to ANALOAF:def 2 :: thesis: verum

end;

assume a,b '||' c,d ; :: thesis: a',b' // c',d'
then ( a,b // c,d or a,b // d,c ) by Def4;
then ( [[a,b],[c,d]] in the CONGR of S or [[a,b],[d,c]] in the CONGR of S ) by ANALOAF:def 2;
hence [[a',b'],[c',d']] in the CONGR of AS by A1, A2, Def1; :: according to ANALOAF:def 2 :: thesis: verum