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 | X) + ((- f2) | X) by Th51
.= (f1 | X) - (f2 | X) by Th53 ; :: thesis: ( (f1 - f2) | X = (f1 | X) - f2 & (f1 - f2) | X = f1 - (f2 | X) )
thus (f1 - f2) | X = (f1 | X) + (- f2) by Th51
.= (f1 | X) - f2 ; :: thesis: (f1 - f2) | X = f1 - (f2 | X)
thus (f1 - f2) | X = f1 + ((- f2) | X) by Th51
.= f1 - (f2 | X) by Th53 ; :: thesis: verum