let X be set ; :: thesis: for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,V 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 V being RealNormSpace
for f1, f2 being PartFunc of C,V holds
( (f1 - f2) | X = (f1 | X) - (f2 | X) & (f1 - f2) | X = (f1 | X) - f2 & (f1 - f2) | X = f1 - (f2 | X) )
let V be RealNormSpace; :: thesis: for f1, f2 being PartFunc of C,V 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,V; :: 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 Th25
.= (f1 | X) + ((- f2) | X) by Th27
.= (f1 | X) + (- (f2 | X)) by Th29
.= (f1 | X) - (f2 | X) by Th25 ; :: thesis: ( (f1 - f2) | X = (f1 | X) - f2 & (f1 - f2) | X = f1 - (f2 | X) )
thus (f1 - f2) | X = (f1 + (- f2)) | X by Th25
.= (f1 | X) + (- f2) by Th27
.= (f1 | X) - f2 by Th25 ; :: thesis: (f1 - f2) | X = f1 - (f2 | X)
thus (f1 - f2) | X = (f1 + (- f2)) | X by Th25
.= f1 + ((- f2) | X) by Th27
.= f1 + (- (f2 | X)) by Th29
.= f1 - (f2 | X) by Th25 ; :: thesis: verum