let X be set ; :: thesis: for C being non empty set
for f1, f2 being PartFunc of C,COMPLEX holds
( (f1 / f2) | X = (f1 | X) / (f2 | X) & (f1 / f2) | X = (f1 | X) / f2 & (f1 / f2) | X = f1 / (f2 | X) )
let C be non empty set ; :: thesis: for f1, f2 being PartFunc of C,COMPLEX holds
( (f1 / f2) | X = (f1 | X) / (f2 | X) & (f1 / f2) | X = (f1 | X) / f2 & (f1 / f2) | X = f1 / (f2 | X) )
let f1, f2 be PartFunc of C,COMPLEX; :: thesis: ( (f1 / f2) | X = (f1 | X) / (f2 | X) & (f1 / f2) | X = (f1 | X) / f2 & (f1 / f2) | X = f1 / (f2 | X) )
thus (f1 / f2) | X = (f1 (#) (f2 ^)) | X by Th38
.= (f1 | X) (#) ((f2 ^) | X) by Th52
.= (f1 | X) (#) ((f2 | X) ^) by Th53
.= (f1 | X) / (f2 | X) by Th38 ; :: thesis: ( (f1 / f2) | X = (f1 | X) / f2 & (f1 / f2) | X = f1 / (f2 | X) )
thus (f1 / f2) | X = (f1 (#) (f2 ^)) | X by Th38
.= (f1 | X) (#) (f2 ^) by Th52
.= (f1 | X) / f2 by Th38 ; :: thesis: (f1 / f2) | X = f1 / (f2 | X)
thus (f1 / f2) | X = (f1 (#) (f2 ^)) | X by Th38
.= f1 (#) ((f2 ^) | X) by Th52
.= f1 (#) ((f2 | X) ^) by Th53
.= f1 / (f2 | X) by Th38 ; :: thesis: verum