Editing Heyting algebra (section)

===Free Heyting algebra on an arbitrary set of generators===
In fact, the preceding construction can be carried out for any set of variables {''A''<sub>''i''</sub> : ''i''∈''I''} (possibly infinite). One obtains in this way the ''free'' Heyting algebra on the variables {''A''<sub>''i''</sub>}, which we will again denote by ''H''<sub>0</sub>. It is free in the sense that given any Heyting algebra ''H'' given together with a family of its elements 〈''a''<sub>''i''</sub>: ''i''∈''I'' 〉, there is a unique morphism ''f'':''H''<sub>0</sub>→''H'' satisfying ''f''([''A''<sub>''i''</sub>])=''a''<sub>''i''</sub>. The uniqueness of ''f'' is not difficult to see, and its existence results essentially from the metaimplication {{nowrap|1 ⇒ 2}} of the section "[[#Provable identities|Provable identities]]" above, in the form of its corollary that whenever ''F'' and ''G'' are provably equivalent formulas, ''F''(〈''a''<sub>''i''</sub>〉)=''G''(〈''a''<sub>''i''</sub>〉) for any family of elements 〈''a''<sub>''i''</sub>〉in ''H''.