let FT be non empty RelStr ; :: thesis: for x being Element of FT

for A being Subset of FT holds

( x in A ^deltai iff ( ex y1, y2 being Element of FT st

( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE ) )

let x be Element of FT; :: thesis: for A being Subset of FT holds

( x in A ^deltai iff ( ex y1, y2 being Element of FT st

( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE ) )

let A be Subset of FT; :: thesis: ( x in A ^deltai iff ( ex y1, y2 being Element of FT st

( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE ) )

A1: ( ex y1, y2 being Element of FT st

( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE implies x in A ^deltai )

( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE ) )

( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE ) ) by A1; :: thesis: verum

for A being Subset of FT holds

( x in A ^deltai iff ( ex y1, y2 being Element of FT st

( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE ) )

let x be Element of FT; :: thesis: for A being Subset of FT holds

( x in A ^deltai iff ( ex y1, y2 being Element of FT st

( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE ) )

let A be Subset of FT; :: thesis: ( x in A ^deltai iff ( ex y1, y2 being Element of FT st

( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE ) )

A1: ( ex y1, y2 being Element of FT st

( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE implies x in A ^deltai )

proof

( x in A ^deltai implies ( ex y1, y2 being Element of FT st
assume
( ex y1, y2 being Element of FT st

( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE ) ; :: thesis: x in A ^deltai

then ( x in A ^delta & x in A ) by Def4, Th8;

hence x in A ^deltai by XBOOLE_0:def 4; :: thesis: verum

end;( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE ) ; :: thesis: x in A ^deltai

then ( x in A ^delta & x in A ) by Def4, Th8;

hence x in A ^deltai by XBOOLE_0:def 4; :: thesis: verum

( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE ) )

proof

hence
( x in A ^deltai iff ( ex y1, y2 being Element of FT st
assume
x in A ^deltai
; :: thesis: ( ex y1, y2 being Element of FT st

( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE )

then ( x in A & x in A ^delta ) by XBOOLE_0:def 4;

hence ( ex y1, y2 being Element of FT st

( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE ) by Def4, Th8; :: thesis: verum

end;( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE )

then ( x in A & x in A ^delta ) by XBOOLE_0:def 4;

hence ( ex y1, y2 being Element of FT st

( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE ) by Def4, Th8; :: thesis: verum

( P_1 (x,y1,A) = TRUE & P_2 (x,y2,A) = TRUE ) & P_A (x,A) = TRUE ) ) by A1; :: thesis: verum