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 Th55:
  for x1,x2,y being FinSequence of COMPLEX st len x1=len x2 & len
  x2=len y holds |((x1+x2),y)| = |(x1,y)| + |(x2,y)|
proof
  let x1,x2,y be FinSequence of COMPLEX;
  assume that
A1: len x1=len x2 and
A2: len x2=len y;
A3: len (Re x1) = len x1 & len (Re x2) = len x2 by Th40;
A4: len (Im y) = len y by Th40;
A5: len (Im x1) = len x1 & len (Im x2) = len x2 by Th40;
A6: len (Re y) = len y by Th40;
  |((x1+x2),y)| = |((Re x1+ Re x2),Re y)| - <i>*(|(Re (x1+x2),Im y)|) +
  <i>*(|(Im (x1+x2),Re y)|) + |(Im (x1+x2),Im y)| by A1,Th41
    .= |((Re x1+ Re x2),Re y)| - <i>*(|((Re x1+Re x2),Im y)|) + <i>*(|(Im (
  x1+x2),Re y)|) + |(Im (x1+x2),Im y)| by A1,Th41
    .= |((Re x1+ Re x2),Re y)| - <i>*(|((Re x1+Re x2),Im y)|) + <i>*(|((Im
  x1+ Im x2),Re y)|) + |(Im (x1+x2),Im y)| by A1,Th41
    .= |((Re x1+ Re x2),Re y)| - <i>*(|((Re x1+Re x2),Im y)|) + <i>*(|((Im
  x1+ Im x2),Re y)|) + |((Im x1+ Im x2),Im y)| by A1,Th41
    .= |(Re x1,Re y)|+ |(Re x2,Re y)| - <i>*(|((Re x1+Re x2),Im y)|) + <i>*(
  |((Im x1+ Im x2),Re y)|) + |((Im x1+ Im x2),Im y)| by A1,A2,A3,A6,RVSUM_1:120
    .= |(Re x1,Re y)|+ |(Re x2,Re y)| - <i>*(|(Re x1,Im y)| + |(Re x2,Im y)|
  ) + <i>*(|((Im x1+ Im x2),Re y)|) + |((Im x1+ Im x2),Im y)| by A1,A2,A3,A4,
RVSUM_1:120
    .= |(Re x1,Re y)|+ |(Re x2,Re y)| - <i>*(|(Re x1,Im y)| + |(Re x2,Im y)|
) + <i>*(|(Im x1,Re y)|+ |(Im x2,Re y)|) + |((Im x1+ Im x2),Im y)| by A1,A2,A6
,A5,RVSUM_1:120
    .= |(Re x1,Re y)| + |(Re x2,Re y)| - <i>*(|(Re x1,Im y)| + |(Re x2,Im y
)|) + <i>*(|(Im x1,Re y)| + |(Im x2,Re y)|) + (|(Im x1,Im y)|+ |(Im x2,Im y)|)
  by A1,A2,A5,A4,RVSUM_1:120
    .= |(x1,y)| + |(x2,y)|;
  hence thesis;
end;
