When we say that two substitutions, say $\theta,\sigma$ agree on the variables of a term $t$ what exactly do we mean ? Is it that both substitution act on the same variables and substitute them with the same terms?
That is $\theta= \lbrace x_1/u_1,x_2/u_2,...,x_n/u_n\rbrace$ and $\sigma= \lbrace x_1/u_1,x_2/u_2,...,x_n/u_n\rbrace$
Or the substitute terms don't really have to be the same? I was confused when i checked the wikipedia lemma about substitution that said :
"Two substitutions are considered equal if they map each variable to structurally equal result terms" link