let P1, P2 be Relation; ( ( for x, y being object holds
( [x,y] in P1 iff ex z being object st
( [x,z] in P & [z,y] in R ) ) ) & ( for x, y being object holds
( [x,y] in P2 iff ex z being object st
( [x,z] in P & [z,y] in R ) ) ) implies P1 = P2 )
assume that
A3:
for x, y being object holds
( [x,y] in P1 iff ex z being object st
( [x,z] in P & [z,y] in R ) )
and
A4:
for x, y being object holds
( [x,y] in P2 iff ex z being object st
( [x,z] in P & [z,y] in R ) )
; P1 = P2
let x be object ; RELAT_1:def 2 for b being object holds
( [x,b] in P1 iff [x,b] in P2 )
let y be object ; ( [x,y] in P1 iff [x,y] in P2 )
( [x,y] in P1 iff ex z being object st
( [x,z] in P & [z,y] in R ) )
by A3;
hence
( [x,y] in P1 iff [x,y] in P2 )
by A4; verum