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 Th51
.= (f1 | X) (#) ((f2 ^ ) | X) by Th65
.= (f1 | X) (#) ((f2 | X) ^ ) by Th66
.= (f1 | X) / (f2 | X) by Th51 ; :: thesis: ( (f1 / f2) | X = (f1 | X) / f2 & (f1 / f2) | X = f1 / (f2 | X) )
thus (f1 / f2) | X = (f1 (#) (f2 ^ )) | X by Th51
.= (f1 | X) (#) (f2 ^ ) by Th65
.= (f1 | X) / f2 by Th51 ; :: thesis: (f1 / f2) | X = f1 / (f2 | X)
thus (f1 / f2) | X = (f1 (#) (f2 ^ )) | X by Th51
.= f1 (#) ((f2 ^ ) | X) by Th65
.= f1 (#) ((f2 | X) ^ ) by Th66
.= f1 / (f2 | X) by Th51 ; :: thesis: verum