let X1, X2 be set ; :: thesis: ( ( for x being object holds
( x in X1 iff ex y, z being object st
( x = [y,z] & [{y},z] in Trace f ) ) ) & ( for x being object holds
( x in X2 iff ex y, z being object st
( x = [y,z] & [{y},z] in Trace f ) ) ) implies X1 = X2 )
assume A1: ( ( for x being object holds
( x in X1 iff ex y, z being object st
( x = [y,z] & [{y},z] in Trace f ) ) ) & ( for x being object holds
( x in X2 iff ex y, z being object st
( x = [y,z] & [{y},z] in Trace f ) ) ) & not X1 = X2 ) ; :: thesis: contradiction
then consider x being object such that
A2: ( ( x in X1 & not x in X2 ) or ( x in X2 & not x in X1 ) ) by TARSKI:2;
( x in X2 iff for y, z being object holds
( not x = [y,z] or not [{y},z] in Trace f ) ) by A1, A2;
hence contradiction by A1; :: thesis: verum