let x, y be FinSequence of COMPLEX ; :: thesis: ( len x = len y implies |((x - y),(x - y))| = (|(x,x)| - (2 * (Re |(x,y)|))) + |(y,y)| )
set z = |(x,y)|;
assume len x = len y ; :: thesis: |((x - y),(x - y))| = (|(x,x)| - (2 * (Re |(x,y)|))) + |(y,y)|
then |((x - y),(x - y))| = ((|(x,x)| - |(x,y)|) - |(y,x)|) + |(y,y)| by Th63
.= ((|(x,x)| - |(x,y)|) - (|(x,y)| *')) + |(y,y)| by Th64
.= (|(x,x)| - (|(x,y)| + (|(x,y)| *'))) + |(y,y)|
.= (|(x,x)| - (2 * (Re |(x,y)|))) + |(y,y)| by Th20 ;
hence |((x - y),(x - y))| = (|(x,x)| - (2 * (Re |(x,y)|))) + |(y,y)| ; :: thesis: verum