reserve D for non empty set,
  i,j,k,l for Nat,
  n for Nat,
  x for set,
  a,b,c,r,r1,r2 for Real,
  p,q for FinSequence of REAL,
  MR,MR1 for Matrix of REAL;

theorem Th6:
  a>0 & a<>1 & b>=0 & c>=0 implies b*c*log(a,b*c) = b*c*log(a,b)+b* c*log(a,c)
proof
  assume that
A1: a>0 and
A2: a<>1 and
A3: b>=0 and
A4: c>=0;
  per cases by A3,A4;
  suppose
    b>0 & c = 0;
    hence thesis;
  end;
  suppose
    b=0 & c = 0;
    hence thesis;
  end;
  suppose
    b=0 & c > 0;
    hence thesis;
  end;
  suppose
    b>0 & c > 0;
    hence b*c*log(a,b*c) = b*c*(log(a,b) + log(a,c)) by A1,A2,POWER:53
      .= b*c*log(a,b)+b*c*log(a,c);
  end;
end;
