defpred S_{1}[ object , object ] means for x1, y1 being object st $1 = [x1,y1] holds

P_{1}[x1,y1,$2];

A2: for x being object st x in [:F_{1}(),F_{2}():] holds

ex z being object st

( z in F_{3}() & S_{1}[x,z] )
_{1}(),F_{2}():],F_{3}() such that

A8: for x being object st x in [:F_{1}(),F_{2}():] holds

S_{1}[x,f . x]
from FUNCT_2:sch 1(A2);

take f ; :: thesis: for x, y being object st x in F_{1}() & y in F_{2}() holds

P_{1}[x,y,f . (x,y)]

let x, y be object ; :: thesis: ( x in F_{1}() & y in F_{2}() implies P_{1}[x,y,f . (x,y)] )

assume ( x in F_{1}() & y in F_{2}() )
; :: thesis: P_{1}[x,y,f . (x,y)]

then [x,y] in [:F_{1}(),F_{2}():]
by ZFMISC_1:def 2;

hence P_{1}[x,y,f . (x,y)]
by A8; :: thesis: verum

P

A2: for x being object st x in [:F

ex z being object st

( z in F

proof

consider f being Function of [:F
let x be object ; :: thesis: ( x in [:F_{1}(),F_{2}():] implies ex z being object st

( z in F_{3}() & S_{1}[x,z] ) )

assume x in [:F_{1}(),F_{2}():]
; :: thesis: ex z being object st

( z in F_{3}() & S_{1}[x,z] )

then consider x1, y1 being object such that

A3: ( x1 in F_{1}() & y1 in F_{2}() )
and

A4: x = [x1,y1] by ZFMISC_1:def 2;

consider z being object such that

A5: z in F_{3}()
and

A6: P_{1}[x1,y1,z]
by A1, A3;

take z ; :: thesis: ( z in F_{3}() & S_{1}[x,z] )

thus z in F_{3}()
by A5; :: thesis: S_{1}[x,z]

let x2, y2 be object ; :: thesis: ( x = [x2,y2] implies P_{1}[x2,y2,z] )

assume A7: x = [x2,y2] ; :: thesis: P_{1}[x2,y2,z]

then x1 = x2 by A4, XTUPLE_0:1;

hence P_{1}[x2,y2,z]
by A4, A6, A7, XTUPLE_0:1; :: thesis: verum

end;( z in F

assume x in [:F

( z in F

then consider x1, y1 being object such that

A3: ( x1 in F

A4: x = [x1,y1] by ZFMISC_1:def 2;

consider z being object such that

A5: z in F

A6: P

take z ; :: thesis: ( z in F

thus z in F

let x2, y2 be object ; :: thesis: ( x = [x2,y2] implies P

assume A7: x = [x2,y2] ; :: thesis: P

then x1 = x2 by A4, XTUPLE_0:1;

hence P

A8: for x being object st x in [:F

S

take f ; :: thesis: for x, y being object st x in F

P

let x, y be object ; :: thesis: ( x in F

assume ( x in F

then [x,y] in [:F

hence P