let X be set ; :: thesis: for f1, f2 being complex-valued Function holds
( (f1 / f2) | X = (f1 | X) / (f2 | X) & (f1 / f2) | X = (f1 | X) / f2 & (f1 / f2) | X = f1 / (f2 | X) )
let f1, f2 be complex-valued Function; :: 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 Th31
.= (f1 | X) (#) ((f2 ^) | X) by Th45
.= (f1 | X) (#) ((f2 | X) ^) by Th46
.= (f1 | X) / (f2 | X) by Th31 ; :: thesis: ( (f1 / f2) | X = (f1 | X) / f2 & (f1 / f2) | X = f1 / (f2 | X) )
thus (f1 / f2) | X = (f1 (#) (f2 ^)) | X by Th31
.= (f1 | X) (#) (f2 ^) by Th45
.= (f1 | X) / f2 by Th31 ; :: thesis: (f1 / f2) | X = f1 / (f2 | X)
thus (f1 / f2) | X = (f1 (#) (f2 ^)) | X by Th31
.= f1 (#) ((f2 ^) | X) by Th45
.= f1 (#) ((f2 | X) ^) by Th46
.= f1 / (f2 | X) by Th31 ; :: thesis: verum