let AFP be AffinPlane; :: thesis: for a, b being Element of
for K being Subset of st a,b // K & not a in K holds
for x being Element of ex y being Element of st
( ( x in K & x = y ) or ( not x in K & ex p, p' being Element of st
( p in K & p' in K & p,a // p',x & p,b // p',y & x,y // K ) ) )
let a, b be Element of ; :: thesis: for K being Subset of st a,b // K & not a in K holds
for x being Element of ex y being Element of st
( ( x in K & x = y ) or ( not x in K & ex p, p' being Element of st
( p in K & p' in K & p,a // p',x & p,b // p',y & x,y // K ) ) )
let K be Subset of ; :: thesis: ( a,b // K & not a in K implies for x being Element of ex y being Element of st
( ( x in K & x = y ) or ( not x in K & ex p, p' being Element of st
( p in K & p' in K & p,a // p',x & p,b // p',y & x,y // K ) ) ) )
assume that
A1: a,b // K and
A2: not a in K ; :: thesis: for x being Element of ex y being Element of st
( ( x in K & x = y ) or ( not x in K & ex p, p' being Element of st
( p in K & p' in K & p,a // p',x & p,b // p',y & x,y // K ) ) )
A3: not b in K by A1, A2, AFF_1:49;
A4: K is being_line by A1, AFF_1:40;
let x be Element of ; :: thesis: ex y being Element of st
( ( x in K & x = y ) or ( not x in K & ex p, p' being Element of st
( p in K & p' in K & p,a // p',x & p,b // p',y & x,y // K ) ) )
consider p, q being Element of such that
A5: p in K and
q in K and
p <> q by A1, AFF_1:31, AFF_1:40;
A6: p <> b by A1, A2, A5, AFF_1:49;

now

set B' = Line p,b;
set A = Line p,a;
assume A7: not x in K ; :: thesis: ex y being Element of st
( ( x in K & x = y ) or ( not x in K & ex p, p' being Element of st
( p in K & p' in K & p,a // p',x & p,b // p',y & x,y // K ) ) )
A8: a in Line p,a by AFF_1:26;
Line p,a is being_line by A2, A5, AFF_1:def 3;
then consider C being Subset of such that
A9: x in C and
A10: Line p,a // C by AFF_1:63;
A11: C is being_line by A10, AFF_1:50;
A12: p in Line p,a by AFF_1:26;
then not Line p,a // K by A2, A5, A8, AFF_1:59;
then not C // K by A10, AFF_1:58;
then consider p' being Element of such that
A13: p' in C and
A14: p' in K by A4, A11, AFF_1:72;
Line p,b is being_line by A6, AFF_1:def 3;
then consider D being Subset of such that
A15: p' in D and
A16: Line p,b // D by AFF_1:63;
A17: b in Line p,b by AFF_1:26;
consider M being Subset of such that
A18: x in M and
A19: K // M by A1, AFF_1:40, AFF_1:63;
A20: M is being_line by A19, AFF_1:50;
A21: p in Line p,b by AFF_1:26;
then A22: not Line p,b // K by A5, A3, A17, AFF_1:59;
A23: not D // M

proof

assume D // M ; :: thesis: contradiction
then Line p,b // M by A16, AFF_1:58;
hence contradiction by A22, A19, AFF_1:58; :: thesis: verum

end;

D is being_line by A16, AFF_1:50;
then consider y being Element of such that
A24: y in D and
A25: y in M by A20, A23, AFF_1:72;
A26: p,b // p',y by A21, A17, A15, A16, A24, AFF_1:53;
A27: x,y // K by A18, A19, A25, AFF_1:54;
p,a // p',x by A12, A8, A9, A10, A13, AFF_1:53;
hence ex y being Element of st
( ( x in K & x = y ) or ( not x in K & ex p, p' being Element of st
( p in K & p' in K & p,a // p',x & p,b // p',y & x,y // K ) ) ) by A5, A7, A14, A27, A26; :: thesis: verum

end;

hence ex y being Element of st
( ( x in K & x = y ) or ( not x in K & ex p, p' being Element of st
( p in K & p' in K & p,a // p',x & p,b // p',y & x,y // K ) ) ) ; :: thesis: verum