let X be ARS ; :: thesis: for x, y being Element of X holds
( x >><< y iff x,y are_convergent_wrt the reduction of X )

let x, y be Element of X; :: thesis: ( x >><< y iff x,y are_convergent_wrt the reduction of X )
set R = the reduction of X;
thus ( x >><< y implies x,y are_convergent_wrt the reduction of X ) :: thesis: ( x,y are_convergent_wrt the reduction of X implies x >><< y )
proof
given z being Element of X such that A1: ( z <=*= x & y =*=> z ) ; :: according to ABSRED_0:def 21 :: thesis: x,y are_convergent_wrt the reduction of X
take z ; :: according to REWRITE1:def 7 :: thesis: ( the reduction of X reduces x,z & the reduction of X reduces y,z )
thus ( the reduction of X reduces x,z & the reduction of X reduces y,z ) by A1; :: thesis: verum
end;
set A = the carrier of X;
F0: field the reduction of X c= the carrier of X \/ the carrier of X by RELSET_1:8;
given a being object such that A2: ( the reduction of X reduces x,a & the reduction of X reduces y,a ) ; :: according to REWRITE1:def 7 :: thesis: x >><< y
per cases ( a in field the reduction of X or not a in field the reduction of X ) ;
suppose a in field the reduction of X ; :: thesis: x >><< y
then reconsider z = a as Element of X by F0;
take z ; :: according to ABSRED_0:def 21 :: thesis: ( x =*=> z & z <=*= y )
thus ( the reduction of X reduces x,z & the reduction of X reduces y,z ) by A2; :: according to ABSRED_0:def 3 :: thesis: verum
end;
suppose not a in field the reduction of X ; :: thesis: x >><< y
then ( a = x & a = y ) by ;
hence x >><< y ; :: thesis: verum
end;
end;