let A be non empty set ; :: thesis: for p, q, r being Element of (RealFunc_Lattice A) holds
( (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (q,r)),p) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (p,q)),r) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (q,p)),r) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (r,p)),q) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (r,q)),p) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (p,r)),q) )
let p, q, r be Element of (RealFunc_Lattice A); :: thesis: ( (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (q,r)),p) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (p,q)),r) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (q,p)),r) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (r,p)),q) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (r,q)),p) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (p,r)),q) )
set s = q "/\" r;
thus A1: (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (q "/\" r) "/\" p by LATTICES:def 2
.= (minfuncreal A) . (((minfuncreal A) . (q,r)),p) ; :: thesis: ( (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (p,q)),r) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (q,p)),r) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (r,p)),q) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (r,q)),p) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (p,r)),q) )
thus (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (p,q)),r) by Th11; :: thesis: ( (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (q,p)),r) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (r,p)),q) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (r,q)),p) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (p,r)),q) )
thus (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = p "/\" (q "/\" r)
.= (q "/\" p) "/\" r by Lm8
.= (minfuncreal A) . (((minfuncreal A) . (q,p)),r) ; :: thesis: ( (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (r,p)),q) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (r,q)),p) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (p,r)),q) )
thus A2: (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = p "/\" (q "/\" r)
.= (r "/\" p) "/\" q by Lm8
.= (minfuncreal A) . (((minfuncreal A) . (r,p)),q) ; :: thesis: ( (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (r,q)),p) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (p,r)),q) )
thus (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (r,q)),p) by A1, Th22; :: thesis: (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (p,r)),q)
thus (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (p,r)),q) by A2, Th22; :: thesis: verum