let i be Nat; :: thesis: for X, Y being finite natural-membered set st X <N< Y & i in card X holds
(Sgm0 (X \/ Y)) . i in X
let X, Y be finite natural-membered set ; :: thesis: ( X <N< Y & i in card X implies (Sgm0 (X \/ Y)) . i in X )
assume that
A1: X <N< Y and
A2: i in card X ; :: thesis: (Sgm0 (X \/ Y)) . i in X
set f = (Sgm0 (X \/ Y)) | (card X);
set f0 = Sgm0 (X \/ Y);
set Z = { v where v is Element of X \/ Y : ex k2 being Nat st
( v = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) } ;
A3: rng (Sgm0 (X \/ Y)) = X \/ Y by Def5;
len (Sgm0 (X \/ Y)) = card (X \/ Y) by Th31;
then A4: card X <= len (Sgm0 (X \/ Y)) by NAT_1:44, XBOOLE_1:7;
then A5: len ((Sgm0 (X \/ Y)) | (card X)) = card X by AFINSQ_1:58;
A6: { v where v is Element of X \/ Y : ex k2 being Nat st
( v = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) } c= rng ((Sgm0 (X \/ Y)) | (card X))

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in { v where v is Element of X \/ Y : ex k2 being Nat st
( v = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) } or y in rng ((Sgm0 (X \/ Y)) | (card X)) )
assume y in { v where v is Element of X \/ Y : ex k2 being Nat st
( v = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) } ; :: thesis: y in rng ((Sgm0 (X \/ Y)) | (card X))
then ex v0 being Element of X \/ Y st
( y = v0 & ex k2 being Nat st
( v0 = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) ) ;
hence y in rng ((Sgm0 (X \/ Y)) | (card X)) by A5, FUNCT_1:def 5; :: thesis: verum

end;

then reconsider Z0 = { v where v is Element of X \/ Y : ex k2 being Nat st
( v = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) } as finite set ;
A7: rng ((Sgm0 (X \/ Y)) | (card X)) c= rng (Sgm0 (X \/ Y)) by RELAT_1:99;
rng ((Sgm0 (X \/ Y)) | (card X)) c= { v where v is Element of X \/ Y : ex k2 being Nat st
( v = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) }

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in rng ((Sgm0 (X \/ Y)) | (card X)) or y in { v where v is Element of X \/ Y : ex k2 being Nat st
( v = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) } )
assume A8: y in rng ((Sgm0 (X \/ Y)) | (card X)) ; :: thesis: y in { v where v is Element of X \/ Y : ex k2 being Nat st
( v = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) }
then consider x being set such that
A9: x in dom ((Sgm0 (X \/ Y)) | (card X)) and
A10: y = ((Sgm0 (X \/ Y)) | (card X)) . x by FUNCT_1:def 5;
reconsider y0 = y as Element of X \/ Y by A7, A8, Def5;
ex k2 being Nat st
( y0 = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) by A5, A9, A10;
hence y in { v where v is Element of X \/ Y : ex k2 being Nat st
( v = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) } ; :: thesis: verum

end;

then A11: rng ((Sgm0 (X \/ Y)) | (card X)) = { v where v is Element of X \/ Y : ex k2 being Nat st
( v = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) } by A6, XBOOLE_0:def 10;
A12: X \/ Y <> {} by A2, CARD_1:47, XBOOLE_1:15;

A13: now

assume that
A14: not { v where v is Element of X \/ Y : ex k2 being Nat st
( v = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) } c= X and
A15: not X c= { v where v is Element of X \/ Y : ex k2 being Nat st
( v = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) } ; :: thesis: contradiction
consider v1 being set such that
A16: v1 in { v where v is Element of X \/ Y : ex k2 being Nat st
( v = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) } and
A17: not v1 in X by A14, TARSKI:def 3;
consider v10 being Element of X \/ Y such that
A18: v1 = v10 and
A19: ex k2 being Nat st
( v10 = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) by A16;
A20: v10 in Y by A12, A17, A18, XBOOLE_0:def 3;
reconsider nv10 = v10 as Nat ;
consider v2 being set such that
A21: v2 in X and
A22: not v2 in { v where v is Element of X \/ Y : ex k2 being Nat st
( v = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) } by A15, TARSKI:def 3;
X c= X \/ Y by XBOOLE_1:7;
then consider x2 being set such that
A23: x2 in dom (Sgm0 (X \/ Y)) and
A24: v2 = (Sgm0 (X \/ Y)) . x2 by A3, A21, FUNCT_1:def 5;
reconsider x20 = x2 as Nat by A23;

now

assume x20 < card X ; :: thesis: contradiction
then A25: x20 in card X by NAT_1:45;
card X <= card (X \/ Y) by NAT_1:44, XBOOLE_1:7;
then card X <= len (Sgm0 (X \/ Y)) by Th31;
then ((Sgm0 (X \/ Y)) | (card X)) . x20 = (Sgm0 (X \/ Y)) . x20 by A25, AFINSQ_1:57;
hence contradiction by A5, A11, A22, A24, A25, FUNCT_1:def 5; :: thesis: verum

end;

then A26: len ((Sgm0 (X \/ Y)) | (card X)) <= x20 by A4, AFINSQ_1:58;
consider k20 being Nat such that
A27: v10 = ((Sgm0 (X \/ Y)) | (card X)) . k20 and
A28: k20 in card X by A19;
A29: ((Sgm0 (X \/ Y)) | (card X)) . k20 = (Sgm0 (X \/ Y)) . k20 by A4, A28, AFINSQ_1:57;
reconsider nv2 = v2 as Nat by A24;
k20 < len ((Sgm0 (X \/ Y)) | (card X)) by A5, A28, NAT_1:45;
then A30: k20 < x20 by A26, XXREAL_0:2;
x20 < len (Sgm0 (X \/ Y)) by A23, NAT_1:45;
then nv10 < nv2 by A27, A24, A30, A29, Def5;
hence contradiction by A1, A21, A20, Def6; :: thesis: verum

end;

(Sgm0 (X \/ Y)) | (card X) is one-to-one by FUNCT_1:84;
then A31: dom ((Sgm0 (X \/ Y)) | (card X)),((Sgm0 (X \/ Y)) | (card X)) .: (dom ((Sgm0 (X \/ Y)) | (card X))) are_equipotent by CARD_1:60;
((Sgm0 (X \/ Y)) | (card X)) .: (dom ((Sgm0 (X \/ Y)) | (card X))) = rng ((Sgm0 (X \/ Y)) | (card X)) by RELAT_1:146;
then A32: card { v where v is Element of X \/ Y : ex k2 being Nat st
( v = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) } = card (len ((Sgm0 (X \/ Y)) | (card X))) by A11, A31, CARD_1:21;
then A33: card { v where v is Element of X \/ Y : ex k2 being Nat st
( v = ((Sgm0 (X \/ Y)) | (card X)) . k2 & k2 in card X ) } = card X by A4, AFINSQ_1:58;

A34: now

per cases ( Z0 c= X or X c= Z0 ) by A13;

case Z0 c= X ; :: thesis: Z0 = X

hence Z0 = X by A4, A32, AFINSQ_1:58, Lm1; :: thesis: verum

end;

case X c= Z0 ; :: thesis: Z0 = X

hence Z0 = X by A33, Lm1; :: thesis: verum

end;

end;

end;

((Sgm0 (X \/ Y)) | (card X)) . i = (Sgm0 (X \/ Y)) . i by A2, A4, AFINSQ_1:57;
hence (Sgm0 (X \/ Y)) . i in X by A2, A5, A11, A34, FUNCT_1:def 5; :: thesis: verum