deffunc H_{1}( object , object , object ) -> set = [:(H . ($2,$3)),(G . ($1,$2)):];

consider M being ManySortedSet of [:I,I,I:] such that

A5: for i, j, k being set st i in I & j in I & k in I holds

M . (i,j,k) = H_{1}(i,j,k)
from ALTCAT_1:sch 3();

take M ; :: thesis: for i, j, k being object st i in I & j in I & k in I holds

M . (i,j,k) = [:(H . (j,k)),(G . (i,j)):]

let i, j, k be object ; :: thesis: ( i in I & j in I & k in I implies M . (i,j,k) = [:(H . (j,k)),(G . (i,j)):] )

thus ( i in I & j in I & k in I implies M . (i,j,k) = [:(H . (j,k)),(G . (i,j)):] ) by A5; :: thesis: verum

consider M being ManySortedSet of [:I,I,I:] such that

A5: for i, j, k being set st i in I & j in I & k in I holds

M . (i,j,k) = H

take M ; :: thesis: for i, j, k being object st i in I & j in I & k in I holds

M . (i,j,k) = [:(H . (j,k)),(G . (i,j)):]

let i, j, k be object ; :: thesis: ( i in I & j in I & k in I implies M . (i,j,k) = [:(H . (j,k)),(G . (i,j)):] )

thus ( i in I & j in I & k in I implies M . (i,j,k) = [:(H . (j,k)),(G . (i,j)):] ) by A5; :: thesis: verum