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;

theorem Th43:
  for x,y being FinSequence of COMPLEX st len x=len y holds
  Re(x - y) = Re x - Re y & Im(x - y) = Im x - Im y
proof
  let x,y be FinSequence of COMPLEX;
  assume
A1: len x=len y;
  then
A2: len (x-y)=len x by Th7;
A3: len x=len (x*') by Def1;
  then
A4: len (x+(x*'))=len x by Th6;
A5: len y=len (y*') by Def1;
  then
A6: len (y+(y*'))=len y by Th6;
  thus Re (x-y)=(1/2)*((x-y)+(x*'-y*')) by A1,Th17
    .=(1/2)*(x*'+(x-y)-y*') by A1,A5,A3,A2,Th31
    .=(1/2)*(x*'+((x-y)-y*')) by A1,A5,A3,A2,Th31
    .=(1/2)*(x*'+(x-(y+y*'))) by A1,A5,Th30
    .=(1/2)*(x+x*'-(y+y*')) by A1,A3,A6,Th31
    .= Re x - Re y by A1,A4,A6,Th36;
A7: len (x-(x*'))=len x by A3,Th7;
A8: len (y-(y*'))=len y by A5,Th7;
  thus Im (x-y)=(-1/2*<i>)*((x-y)-(x*'-y*')) by A1,Th17
    .=(-1/2*<i>)*((x-y)-x*'+y*') by A1,A5,A3,A2,Th33
    .=(-1/2*<i>)*(x-(x*'+y)+y*') by A1,A3,Th30
    .=(-1/2*<i>)*(x-x*'-y+y*') by A1,A3,Th30
    .=(-1/2*<i>)*((x-x*')-(y-y*')) by A1,A5,A7,Th33
    .=Im x - Im y by A1,A7,A8,Th36;
end;
