let POS be non empty ParOrtStr ; :: thesis: for a, b, c, d being Element of POS
for a9, b9, c9, d9 being Element of AffinStruct(# the carrier of POS, the CONGR of POS #) st a = a9 & b = b9 & c = c9 & d = d9 holds
( a,b // c,d iff a9,b9 // c9,d9 )
set AF = AffinStruct(# the carrier of POS, the CONGR of POS #);
let a, b, c, d be Element of POS; :: thesis: for a9, b9, c9, d9 being Element of AffinStruct(# the carrier of POS, the CONGR of POS #) st a = a9 & b = b9 & c = c9 & d = d9 holds
( a,b // c,d iff a9,b9 // c9,d9 )
let a9, b9, c9, d9 be Element of AffinStruct(# the carrier of POS, the CONGR of POS #); :: thesis: ( a = a9 & b = b9 & c = c9 & d = d9 implies ( a,b // c,d iff a9,b9 // c9,d9 ) )
assume A1: ( a = a9 & b = b9 & c = c9 & d = d9 ) ; :: thesis: ( a,b // c,d iff a9,b9 // c9,d9 )

hereby :: thesis: ( a9,b9 // c9,d9 implies a,b // c,d )

assume a,b // c,d ; :: thesis: a9,b9 // c9,d9
then [[a9,b9],[c9,d9]] in the CONGR of AffinStruct(# the carrier of POS, the CONGR of POS #) by A1, ANALOAF:def 2;
hence a9,b9 // c9,d9 by ANALOAF:def 2; :: thesis: verum

end;

assume a9,b9 // c9,d9 ; :: thesis: a,b // c,d
then [[a,b],[c,d]] in the CONGR of POS by A1, ANALOAF:def 2;
hence a,b // c,d by ANALOAF:def 2; :: thesis: verum