A3:
for n being Nat
for x being Element of F1() ex y being Element of F1() st P1[n,x,y]
by A1;
consider f being sequence of F1() such that
A4:
( f . 0 = F2() & ( for n being Nat holds P1[n,f . n,f . (n + 1)] ) )
from RECDEF_1:sch 2(A3);
A5:
for n being Nat
for x, y1, y2 being Element of F1() st P1[n,x,y1] & P1[n,x,y2] holds
y1 = y2
by A2;
thus
ex y being Element of F1() ex f being sequence of F1() st
( y = f . F3() & f . 0 = F2() & ( for n being Nat holds P1[n,f . n,f . (n + 1)] ) )
for y1, y2 being Element of F1() st ex f being sequence of F1() st
( y1 = f . F3() & f . 0 = F2() & ( for n being Nat holds P1[n,f . n,f . (n + 1)] ) ) & ex f being sequence of F1() st
( y2 = f . F3() & f . 0 = F2() & ( for n being Nat holds P1[n,f . n,f . (n + 1)] ) ) holds
y1 = y2proof
reconsider n =
F3() as
Element of
NAT by ORDINAL1:def 12;
take
f . n
;
ex f being sequence of F1() st
( f . n = f . F3() & f . 0 = F2() & ( for n being Nat holds P1[n,f . n,f . (n + 1)] ) )
take
f
;
( f . n = f . F3() & f . 0 = F2() & ( for n being Nat holds P1[n,f . n,f . (n + 1)] ) )
thus
(
f . n = f . F3() &
f . 0 = F2() & ( for
n being
Nat holds
P1[
n,
f . n,
f . (n + 1)] ) )
by A4;
verum
end;
let y1, y2 be Element of F1(); ( ex f being sequence of F1() st
( y1 = f . F3() & f . 0 = F2() & ( for n being Nat holds P1[n,f . n,f . (n + 1)] ) ) & ex f being sequence of F1() st
( y2 = f . F3() & f . 0 = F2() & ( for n being Nat holds P1[n,f . n,f . (n + 1)] ) ) implies y1 = y2 )
given f1 being sequence of F1() such that A6:
y1 = f1 . F3()
and
A7:
f1 . 0 = F2()
and
A8:
for n being Nat holds P1[n,f1 . n,f1 . (n + 1)]
; ( for f being sequence of F1() holds
( not y2 = f . F3() or not f . 0 = F2() or ex n being Nat st P1[n,f . n,f . (n + 1)] ) or y1 = y2 )
A9:
f1 . 0 = F2()
by A7;
A10:
for n being Nat holds P1[n,f1 . n,f1 . (n + 1)]
by A8;
given f2 being sequence of F1() such that A11:
y2 = f2 . F3()
and
A12:
f2 . 0 = F2()
and
A13:
for n being Nat holds P1[n,f2 . n,f2 . (n + 1)]
; y1 = y2
A14:
for n being Nat holds P1[n,f2 . n,f2 . (n + 1)]
by A13;
A15:
f2 . 0 = F2()
by A12;
f1 = f2
from NAT_1:sch 14(A9, A10, A15, A14, A5);
hence
y1 = y2
by A6, A11; verum