A3: for n being Element of NAT
for x being Element of F1() ex y being Element of F1() st P1[n,x,y] by A1;
consider f being Function of NAT ,F1() such that
A4: ( f . 0 = F2() & ( for n being Element of 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
proof
let n be Nat; :: thesis: for x, y1, y2 being Element of F1() st P1[n,x,y1] & P1[n,x,y2] holds
y1 = y2
n in NAT by ORDINAL1:def 13;
hence for x, y1, y2 being Element of F1() st P1[n,x,y1] & P1[n,x,y2] holds
y1 = y2 by A2; :: thesis: verum
end;
thus ex y being Element of F1() ex f being Function of NAT ,F1() st
( y = f . F3() & f . 0 = F2() & ( for n being Element of NAT holds P1[n,f . n,f . (n + 1)] ) ) :: thesis: for y1, y2 being Element of F1() st ex f being Function of NAT ,F1() st
( y1 = f . F3() & f . 0 = F2() & ( for n being Element of NAT holds P1[n,f . n,f . (n + 1)] ) ) & ex f being Function of NAT ,F1() st
( y2 = f . F3() & f . 0 = F2() & ( for n being Element of NAT holds P1[n,f . n,f . (n + 1)] ) ) holds
y1 = y2
proof
reconsider n = F3() as Element of NAT by ORDINAL1:def 13;
take f . n ; :: thesis: ex f being Function of NAT ,F1() st
( f . n = f . F3() & f . 0 = F2() & ( for n being Element of NAT holds P1[n,f . n,f . (n + 1)] ) )
take f ; :: thesis: ( f . n = f . F3() & f . 0 = F2() & ( for n being Element of NAT holds P1[n,f . n,f . (n + 1)] ) )
thus ( f . n = f . F3() & f . 0 = F2() & ( for n being Element of NAT holds P1[n,f . n,f . (n + 1)] ) ) by A4; :: thesis: verum
end;
let y1, y2 be Element of F1(); :: thesis: ( ex f being Function of NAT ,F1() st
( y1 = f . F3() & f . 0 = F2() & ( for n being Element of NAT holds P1[n,f . n,f . (n + 1)] ) ) & ex f being Function of NAT ,F1() st
( y2 = f . F3() & f . 0 = F2() & ( for n being Element of NAT holds P1[n,f . n,f . (n + 1)] ) ) implies y1 = y2 )
given f1 being Function of NAT ,F1() such that A6: y1 = f1 . F3() and
A7: f1 . 0 = F2() and
A8: for n being Element of NAT holds P1[n,f1 . n,f1 . (n + 1)] ; :: thesis: ( for f being Function of NAT ,F1() holds
( not y2 = f . F3() or not f . 0 = F2() or ex n being Element of 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)]
proof
let n be Nat; :: thesis: P1[n,f1 . n,f1 . (n + 1)]
n in NAT by ORDINAL1:def 13;
hence P1[n,f1 . n,f1 . (n + 1)] by A8; :: thesis: verum
end;
given f2 being Function of NAT ,F1() such that A11: y2 = f2 . F3() and
A12: f2 . 0 = F2() and
A13: for n being Element of NAT holds P1[n,f2 . n,f2 . (n + 1)] ; :: thesis: y1 = y2
A14: for n being Nat holds P1[n,f2 . n,f2 . (n + 1)]
proof
let n be Nat; :: thesis: P1[n,f2 . n,f2 . (n + 1)]
n in NAT by ORDINAL1:def 13;
hence P1[n,f2 . n,f2 . (n + 1)] by A13; :: thesis: verum
end;
A15: f2 . 0 = F2() by A12;
f1 = f2 from NAT_1:sch 14(A9, A10, A15, A14, A5);
 hence y1 = y2 by A6, A11; :: thesis: verum