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 Th45:
  for c being Complex holds (-c)*x = -(c*x)
proof
  let c be Complex;
A1: len ((-c)*x)=len x by Th3;
A2: len (c*x)=len x by Th3;
  (-c)*x + c*x = (-1)*c*x + c*x .= -(c*x)+ c*x by Th44
    .= -(c*x - c*x) by A2,Th32
    .= -(0c (len x)) by A2,Th28
    .= 0c (len x) by Lm3;
  hence thesis by A1,A2,Th26;
end;
