A3:
for c being set holds
( not c in F1() or P1[c] or P2[c] or P3[c] )
by A2;
A4:
for c being set st c in F1() holds
( ( P1[c] implies not P2[c] ) & ( P1[c] implies not P3[c] ) & ( P2[c] implies not P3[c] ) )
by A1;
ex f being Function st
( dom f = F1() & ( for c being set st c in F1() holds
( ( P1[c] implies f . c = F2(c) ) & ( P2[c] implies f . c = F3(c) ) & ( P3[c] implies f . c = F4(c) ) ) ) )
from RECDEF_2:sch 1(A4, A3);
then consider f being Function such that
A5:
dom f = F1()
and
A6:
for c being set st c in F1() holds
( ( P1[c] implies f . c = F2(c) ) & ( P2[c] implies f . c = F3(c) ) & ( P3[c] implies f . c = F4(c) ) )
;
take
f
; ( dom f = F1() & ( for c being Element of F1() holds
( ( P1[c] implies f . c = F2(c) ) & ( P2[c] implies f . c = F3(c) ) & ( P3[c] implies f . c = F4(c) ) ) ) )
thus
dom f = F1()
by A5; for c being Element of F1() holds
( ( P1[c] implies f . c = F2(c) ) & ( P2[c] implies f . c = F3(c) ) & ( P3[c] implies f . c = F4(c) ) )
let c be Element of F1(); ( ( P1[c] implies f . c = F2(c) ) & ( P2[c] implies f . c = F3(c) ) & ( P3[c] implies f . c = F4(c) ) )
thus
( ( P1[c] implies f . c = F2(c) ) & ( P2[c] implies f . c = F3(c) ) & ( P3[c] implies f . c = F4(c) ) )
by A6; verum