reserve i,j for Element of NAT,
  x,y,z for FinSequence of COMPLEX,
  c for Element of COMPLEX,
  R,R1,R2 for Element of i-tuples_on COMPLEX;
reserve C for Function of [:COMPLEX,COMPLEX:],COMPLEX;
reserve G for Function of [:REAL,REAL:],REAL;
reserve h for Function of COMPLEX,COMPLEX,
  g for Function of REAL,REAL;

theorem Th61:
  for x,y1,y2 being FinSequence of COMPLEX st len x=len y1 & len y1=len y2
  holds |(x, y1-y2)| = |(x, y1)| - |(x, y2)|
proof
  let x,y1,y2 be FinSequence of COMPLEX;
  assume that
A1: len x=len y1 and
A2: len y1=len y2;
A3: len (Re y1) = len y1 & len (Re y2) = len y2 by Th40;
A4: len (Im x) = len x by Th40;
A5: len (Im y1) = len y1 & len (Im y2) = len y2 by Th40;
A6: len (Re x) = len x by Th40;
  |(x,(y1-y2))| = |(Re x,(Re y1- Re y2))| - <i>*(|(Re x,Im (y1-y2))|) +
  <i>*(|(Im x,Re (y1-y2))|) + |(Im x,Im (y1-y2))| by A2,Th43
    .= |(Re x,(Re y1- Re y2))| - <i>*(|(Re x,Im (y1-y2))|) + <i>*(|(Im x,Re
  (y1-y2))|) + |(Im x,(Im y1-Im y2))| by A2,Th43
    .= |(Re x,(Re y1- Re y2))| - <i>*(|(Re x,(Im y1-Im y2))|) + <i>*(|(Im x,
  Re (y1-y2))|) + |(Im x,(Im y1-Im y2))| by A2,Th43
    .= |(Re x,(Re y1- Re y2))| - <i>*(|(Re x,(Im y1-Im y2))|) + <i>*(|(Im x,
  (Re y1-Re y2))|) + |(Im x,(Im y1-Im y2))| by A2,Th43
    .= |(Re x,Re y1)|- |(Re x,Re y2)| - <i>*(|(Re x,(Im y1-Im y2))|) + <i>*(
  |(Im x,(Re y1-Re y2))|) + |(Im x,(Im y1-Im y2))| by A1,A2,A3,A6,RVSUM_1:124
    .= |(Re x,Re y1)|- |(Re x,Re y2)| - <i>*(|(Re x,(Im y1-Im y2))|) + <i>*(
  |(Im x,(Re y1-Re y2))|) + (|(Im x,Im y1)|-|(Im x,Im y2)|) by A1,A2,A5,A4,
RVSUM_1:124
    .= |(Re x,Re y1)|- |(Re x,Re y2)| - <i>*(|(Re x,Im y1)|-|(Re x,Im y2)|)
+ <i>*(|(Im x,(Re y1-Re y2))|) + (|(Im x,Im y1)|-|(Im x,Im y2)|) by A1,A2,A6,A5
,RVSUM_1:124
    .= |(Re x,Re y1)|- |(Re x,Re y2)| - <i>*(|(Re x,Im y1)|-|(Re x,Im y2)|)
+ <i>*(|(Im x,Re y1)|-|(Im x,Re y2)|) + (|(Im x,Im y1)|-|(Im x,Im y2)|) by A1
,A2,A3,A4,RVSUM_1:124
    .= |(x,y1)| - |(x,y2)|;
  hence thesis;
end;
