let X be ARS ; :: thesis: for x, y, z being Element of X st x =01=> y & y =01=> z holds
x =*=> z
let x, y, z be Element of X; :: thesis: ( x =01=> y & y =01=> z implies x =*=> z )
assume A1: x =01=> y ; :: thesis: ( not y =01=> z or x =*=> z )
assume A2: y =01=> z ; :: thesis: x =*=> z
A3: x =*=> y by A1, Lem1;
A4: y =*=> z by A2, Lem1;
thus x =*=> z by A3, A4, Th3; :: thesis: verum