let a, b, c, d be Element of AFV0; :: thesis: for a', b', c', d' being Element of the carrier of AffinStruct(# the carrier of AFV0,P #) 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 the carrier of AffinStruct(# the carrier of AFV0,P #); :: thesis: ( a = a' & b = b' & c = c' & d = d' implies ( a,b '||' c,d iff a',b' // c',d' ) )
assume A1: ( a = a' & b = b' & c = c' & d = d' ) ; :: thesis: ( a,b '||' c,d iff a',b' // c',d' )

A2: now

assume a,b '||' c,d ; :: thesis: a',b' // c',d'
then [[a,b],[c,d]] in the CONGR of AffinStruct(# the carrier of AFV0,P #) by Lm14;
hence a',b' // c',d' by A1, ANALOAF:def 2; :: thesis: verum

end;

now

assume a',b' // c',d' ; :: thesis: a,b '||' c,d
then [[a',b'],[c',d']] in P by ANALOAF:def 2;
hence a,b '||' c,d by A1, Lm14; :: thesis: verum

end;

hence ( a,b '||' c,d iff a',b' // c',d' ) by A2; :: thesis: verum