let X be ARS ; :: thesis: for x, y, z being Element of X st x ==> y & y ==> z holds
x =*=> z
let x, y, z be Element of X; :: thesis: ( x ==> y & y ==> z implies x =*=> z )
assume A1: x ==> y ; :: thesis: ( not y ==> z or x =*=> z )
assume A2: y ==> z ; :: thesis: x =*=> z
A3: x =*=> y by A1, Th2;
A4: y =*=> z by A2, Th2;
thus x =*=> z by A3, A4, Th3; :: thesis: verum