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 | X) + ((- f2) | X) by Th44
.= (f1 | X) - (f2 | X) by Th46 ; :: thesis: ( (f1 - f2) | X = (f1 | X) - f2 & (f1 - f2) | X = f1 - (f2 | X) )
thus (f1 - f2) | X = (f1 | X) - f2 by Th44; :: thesis: (f1 - f2) | X = f1 - (f2 | X)
thus (f1 - f2) | X = f1 + ((- f2) | X) by Th44
.= f1 - (f2 | X) by Th46 ; :: thesis: verum