reserve x,y,z,w for ExtReal,
  r for Real;
reserve f,g for ExtReal;
reserve t for ExtReal;

theorem :: MEMBER_1:6
  r <> 0 & r*f = r*g implies f = g
proof
  assume that
A1: r <> 0 and
A2: r*f = r*g;
A3: r is positive or r is negative by A1;
  per cases by XXREAL_0:14;
  suppose
A4: f in REAL;
    now
      assume not g in REAL;
      then g = +infty or g = -infty by XXREAL_0:14;
      hence contradiction by A2,A3,A4,Def5;
    end;
    then
A5: ex c, d being Complex st r = c & g = d & r*g = c*d by Def5;
    ex a, b being Complex st r = a & f = b & r*f = a *b by A4,Def5;
    hence thesis by A1,A2,A5,XCMPLX_1:5;
  end;
  suppose
A6: f = +infty;
    assume f <> g;
    then g in REAL or g = -infty by A6,XXREAL_0:14;
    hence thesis by A2,A3,A6,Def5;
  end;
  suppose
A7: f = -infty;
    assume f <> g;
    then g in REAL or g = +infty by A7,XXREAL_0:14;
    hence thesis by A2,A3,A7,Def5;
  end;
end;
