let P1, P2 be Relation; :: thesis: ( ( 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 ) ) ; :: thesis: P1 = P2
let x be object ; :: according to RELAT_1:def 2 :: thesis: for b being object holds
( [x,b] in P1 iff [x,b] in P2 )
let y be object ; :: thesis: ( [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; :: thesis: verum