let AFP be AffinPlane; :: thesis: for a, b being Element of AFP
for K being Subset of AFP
for f being Permutation of the carrier of AFP st a,b // K & not a in K & ( for x, y being Element of AFP holds
( f . x = y iff ( ( x in K & x = y ) or ( not x in K & ex p, p9 being Element of AFP st
( p in K & p9 in K & p,a // p9,x & p,b // p9,y & x,y // K ) ) ) ) ) holds
f . a = b
let a, b be Element of AFP; :: thesis: for K being Subset of AFP
for f being Permutation of the carrier of AFP st a,b // K & not a in K & ( for x, y being Element of AFP holds
( f . x = y iff ( ( x in K & x = y ) or ( not x in K & ex p, p9 being Element of AFP st
( p in K & p9 in K & p,a // p9,x & p,b // p9,y & x,y // K ) ) ) ) ) holds
f . a = b
let K be Subset of AFP; :: thesis: for f being Permutation of the carrier of AFP st a,b // K & not a in K & ( for x, y being Element of AFP holds
( f . x = y iff ( ( x in K & x = y ) or ( not x in K & ex p, p9 being Element of AFP st
( p in K & p9 in K & p,a // p9,x & p,b // p9,y & x,y // K ) ) ) ) ) holds
f . a = b
let f be Permutation of the carrier of AFP; :: thesis: ( a,b // K & not a in K & ( for x, y being Element of AFP holds
( f . x = y iff ( ( x in K & x = y ) or ( not x in K & ex p, p9 being Element of AFP st
( p in K & p9 in K & p,a // p9,x & p,b // p9,y & x,y // K ) ) ) ) ) implies f . a = b )
assume that
A1: a,b // K and
A2: not a in K ; :: thesis: ( ex x, y being Element of AFP st
( ( not f . x = y or ( x in K & x = y ) or ( not x in K & ex p, p9 being Element of AFP st
( p in K & p9 in K & p,a // p9,x & p,b // p9,y & x,y // K ) ) ) implies ( ( ( x in K & x = y ) or ( not x in K & ex p, p9 being Element of AFP st
( p in K & p9 in K & p,a // p9,x & p,b // p9,y & x,y // K ) ) ) & not f . x = y ) ) or f . a = b )
consider p, q being Element of AFP such that
A3: p in K and
q in K and
p <> q by A1, AFF_1:19, AFF_1:26;
A4: p,b // p,b by AFF_1:2;
assume A5: for x, y being Element of AFP holds
( f . x = y iff ( ( x in K & x = y ) or ( not x in K & ex p, p9 being Element of AFP st
( p in K & p9 in K & p,a // p9,x & p,b // p9,y & x,y // K ) ) ) ) ; :: thesis: f . a = b
p,a // p,a by AFF_1:2;
hence f . a = b by A1, A2, A5, A3, A4; :: thesis: verum